THEORY  OF  ERRORS  AND 
LEAST  SQUARES 


THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON   •    CHICAGO  •    DALLAS 
ATLANTA   •    SAN   FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON  •    BOMBAY  •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


THEORY  OF  ERRORS  AND 
LEAST  SQUARES 


A  TEXTBOOK  FOR  COLLEGE  STUDENTS 
AND  RESEARCH  WORKERS 


BY 
LeROY   D.    WELD,   M.S. 

PROFESSOR   OF   PHYSICS   IN   COE   COLLEGE 


Neto  gork 

THE   MACMILLAN   COMPANY 

1922 

All  rights  reserved 


9  7sr 


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COPYKIGHT,    1916, 

By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  March,  1916. 

Eclucair^on     OtpT- 


NorbjooD  iPress 

J.  8.  CushinK  Co.  -  lieiwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

There  are  few  branches  of  mathematics  which  have 
wider  applicability  to  general  scientific  work  than  the 
Theory  of  Errors,  and  few  mathematical  implements 
which  are  capable  of  greater  usefulness  to  the  research 
worker  than  the  Method  of  Least  Squares.  Yet,  for 
some  reason,  students  are  rarely  given  opportunity  to 
acquire  facility  in  these  lines,  the  result  being  that  too 
many  of  our  scientists  and  engineers  go  about  their 
work  without  such  equipment.  It  would  be  almost 
impossible  to  enumerate  the  variety  of  ways  in  which 
the  ideas  relating  to  these  subjects  adapt  themselves 
to  even  such  simple  bits  of  quantitative  work  as  the 
chemist  or  the  surveyor  is  daily  called  upon  to  do. 
And  it  is  difficult  for  the  writer  to  imagine  how  an 
elaborate  research  in  any  of  the  exact  sciences  can 
be  carried  on  at  all,  without  the  constant  application 
of  these  principles  throughout  both  the  preliminary 
and  the  final  stages  of  the  work.  The  satisfaction  to 
be  gained  from  the  application  of  the  theory  of  pre- 
cision alone  is  well  worth  all  the  time  necessary  to 
acquire  these  subjects.     Add  to  this  the  fact  that  the 

5004'.  a 


VI  PREFACE 

theory  of  error  distribution  has  direct  theoretical  bear- 
ing upon  certain  very  important  laws  and  problems  of 
physics,  chemistry,  astronomy,  and  even  of  biology, 
and  the  reasons  for  students'  having  opportunity  to 
attain  the  elements  of  the  subject  become  still  more 
emphatic. 

This  small  volume  embodies  the  material  used  by 
the  writer  as  lecture  notes  during  the  past  twelve 
years.  It  is  intended  as  a  presentation  of  the  Theory 
of  Errors  and  Least  Squares  in  such  a  simple  and  con- 
cise form  as  to  be  useful,  not  only  as  a  textbook  for 
undergraduates,  but  as  a  handy  reference  which  any 
research  worker  can  read  through  in  an  evening  or  so 
and  then  put  into  immediate  practice. 

It  will  be  noticed  that  the  illustrative  examples  and 
problems  are  drawn  from  various  branches  of  science, 
suggesting  the  wide  range  of  possible  application.  No 
attempt  is  made,  of  course,  at  an  exhaustive  treatment 
in  such  small  compass.  Some  of  the  special  methods 
employed  by  expert  computers,  often  included  in  larger 
works,  have  been  purposely  omitted.  For  the  conven- 
ience of  the  student,  and  in  order  not  to  interrupt  the 
thread  of  the  subject,  a  few  of  the  more  complicated 
mathematical  discussions  have  been  set  apart  in  the 
Appendix  and  referred  to  at  the  appropriate  places. 
It  is  not  intended  that  they  shall  be  omitted  from  the 
course  when  using  the  book  as  a  text,  though  the  cas- 
ual reader  may  get  along  ver}^  well  without  them. 

The  writer  wishes  to  express  his  appreciation  to  the 
numerous  friends  who  have  kindly  given  aid  by  way 


PREFACE  VU 

of  furnishing  data  for  the  illustrative  examples,  or 
otherwise.  Where  material  has  been  taken  from  other 
works,  due  credit  has  been  given  for  the  same. 

L.  D.  W. 

Cedar  Rapids,  Iowa, 
December,  1915. 


CONTENTS 

CHAPTER  I 
ON  MEASUREMENT 

ARTICLE  PAOB 

1.  Definition  of  measurement 1 

2.  Indirect  measurement 1 

3.  Estimation 3 

4.  The  impossibility  of  exact  measurements         ...  5 

5.  Errors  of  measurement 6 

6.  Exercises .        .  8 

CHAPTER  II 

ON  THE  OCCURRENCE  AND  GENERAL 
PROPERTIES  OF  ERRORS 

7.  Errors  and  residuals •        ,11 

8.  Classification  of  errors .  13 

9.  Mistakes ,17 

10.  General  methods  of  eliminating  persistent  errors     .        .  18 

11.  Exercises  leading  to  an  understanding  of  error  distri- 

bution           22 

12.  Remarks  on  the  distribution  of  errors       ....  26 

13.  Precision 28 

14.  Mathematical  expression  of  the  law  of  error    .        .        .  30 


CHAPTER  III 
ON  PROBABILITIES 

15.  Fundamental  principle       .         .         ...        .        .        .31 

16.  Definition  of  mathematical  probability     ....      32 

17.  Permutations .34 


X  CONTENTS 

ARTICLE  PAGE 

18.  Combinations 36 

19.  Probability  of  either  of  two  or  more  events      ...  39 

20.  Probability  of  the  concurrence  of  independent  events      .  40 

21.  The  coin  problem 41 

22.  Important  exercise 44 

23.  Empirical  or  statistical  probability 45 

24.  Exercises 45 


CHAPTER  IV 

THE  ERROR  EQUATION  AND  THE  PRINCIPLE 
OF   LEAST   SQUARES 

25.  Analogy  of  error  distribution  to  coin  problem 

26.  The  most  probable  value  from  a  series  of  direct  measure- 

ments.    The  arithmetical  mean 

27.  Gauss's  deduction  of  the  error  equation    . 

28.  Discussion  of  the  error  equation 

29.  The  principle  of  least  squares  in  its  simplest  form 

30.  Exercises 


51 
52 

56 

58 
61 


CHAPTER  V 

ON  THE  ADJUSTMENT  OF  INDIRECT 
OBSERVATIONS 

31.  Observations  on  functions  of  a  single  quantity         .        .  65 

32.  Observation    equations   for    more    than    one    unknown 

quantity 67 

33.  More  observations  than  quantities.     Normal  equations   .  69 

34.  Reduction  of  observation  equations  of  the  first  degree     .  72 

35.  Illustrations  from  physics 74 

36.  Illustrations  from  chemistry 78 

37.  Illustrations  from  surveying 81 

38.  Illustrations  from  astronomy 85 

39.  Observation  equations  not  of  first  degree  ...  89 

40.  Observations  upon   quantities  subject  to  rigorous  con- 

ditions           91 

41.  Exercises 94 


CONTENTS 


XI 


CHAPTER  VI 
EMPIRICAL   FORMULAS 


42.  Classification  of  formulas 

43.  Uses  and  limitations  of  empirical  formulas 

44.  Illustrations  of  empirical  formulas   . 

45.  Choice  of  mathematical  expression  . 

46.  Exercises 


PAGE 

104 
105 
107 
111 
115 


CHAPTER   VII 
WEIGHTED  OBSERVATIONS 

47.  Relative  reliability  of  observations.     Weights  .        .  124 

48.  Adjustment  of  observations  of  unequal  weight         .         .  126 

49.  Exercises 128 

50.  Wild  or  doubtful  observations 135 

51.  The  precision  index  h 136 

52.  General  statement  of  the  principle  of  least  squares  .        .  139 


CHAPTER  VIII 
PRECISION  AND  THE   PROBABLE   ERROR 

53.  Discontinuity  of  the  error  variable 

54.  Value  of  the  integral  /,  and  the  relation  between  c  and  h 

55.  Probability  of  an  error  lying  between  given  limits.     The 

probability  integral 

56.  Calculation  of  the  precision  index  from  the  residuals 

57.  Approximate  formulas  for  the  precision  index 

58.  The  probable  error  of  an  observation 

59.  Relation  between  probable  error  and  weight    . 

60.  Exercises 

61.  Probable  errors  of  functions  of  observed  quantities 

62.  Probable  errors  of  adjusted  values     .... 

63.  Probable  errors  of  conditioned  observations     . 

64.  Exercises 


141 

143 

144 
146 
149 
152 
155 
159 
163 
166 
170 
171 


Xll  CONTENTS 

APPENDIX 
SUPPLEMENTARY  NOTES 

PAGE 

A.  Proof  of  the  necessary  functional  relation  assumed  in 

deriving  the  error  law.    (Supplementary  to  Art.  27)     177 

B.  Approximation  method  for  observation  equations  not  of 

the  first  degree.     (Supplementary  to  Art.  39)  .         .     178 

/.OO 

C.  Evaluation  of  the  integral  |    e-^^^dx.       (Supplementary 

to  Art.  54) 180 

Z).   Evaluation  of  the  probability  integral.     (Supplementary 

to  Art.  55) 182 

E.  Outline  of  another  method  for  probable  errors  of  adjusted 

values.     (Supplementary  to  Art.  62)         .         .         .     183 

F.  Collection  of  important  definitions,  theorems,  rules,  and 

formulas  for  convenient  reference      ....     185 


THEORY  OF  ERRORS  AND 
LEAST  SQUARES 


THEORY    OF   ERRORS   AND   LEAST 
SQUARES 

CHAPTER  I 
ON   MEASUREMENT 

1.  Definition  of  Measurement.  —  To  measure  a  quan- 
tity is  to  determine  by  any  means,  direct  or  indirect,  its 
ratio  to  the  unit  employed  in  expressing  the  value  of  that 
quantity.  Thus,  in  measuring  a  Hue,  we  find  that  it  is 
a  certain  number  of  times  as  long  as  the  foot  or  the  centi- 
meter, and  this  number  is  said  to  be  its  value  in  feet  or 
centimeters. 

This  definition  must  be  clearly  understood  to  be  in- 
dependent of  whatever  process  is  used  in  the  measurement. 
We  could  measure  the  area  of  a  polygonal  piece  of  sheet 
iron  in  two  ways  :  either  by  measuring  its  sides  and  angles 
and  computing  its  area  by  geometry,  or  by  weighing  it 
and  comparing  its  weight  to  that  of  a  square  piece  with 
unit  side.  Either  of  these  processes  is  a  true  measure- 
ment of  the  area,  though  neither  is  a  direct  measurement. 

2.  Indirect  Measurement.  —  Indeed,  with  the  excep- 
tion of  one  kind  of  magnitude,  very  few  measurements 
are  direct.     By  this  is  meant  that  we  do  not,  in  general, 

B  1 


;2      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

apply  the  unit  of  measure  directly  to  the  magnitude  to 
be  measured.  This  is  done  commonly  only  in  the  case  of 
length.  We  can,  in  measuring  a  line,  apply  the  yardstick 
directly  along  the  line  and  determine  in  this  way  how 
many  times  greater  one  is  than  the  other.  But  we  cannot 
take  a  lamp  in  one  hand  and  a  standard  candle  in  the 
other  and  determine  the  candle-power  of  the  lamp  in 
any  such  direct  manner. 

So  far  is  the  above  mentioned  principle  true,  that,  as  a 
matter  of  fact,  nearly  every  kind  of  measurement  is 
made  to  depend,  in  practice,  upon  measurements  of  length. 
This  will  be  clear  from  a  number  of  illustrations. 

Angles  are  measured,  not  by  applying  the  wedge-like 
degree  as  a  unit,  but  by  measuring  the  length  of  the  arc 
laid  off  on  a  curved  linear  scale,  or  by  measuring  the 
lengths  of  straight  lines  connected  with  the  angle  and  com- 
puting the  latter  from  its  trigonometric  functions. 

Time  is  measured,  not  by  counting  the  minutes  and 
seconds  in  the  interval,  but  by  observing  the  motion  of 
the  clock  hands  over  a  curvilinear  scale  called  the  dial, 
marked  off  in  spaces  of  equal  length;  or  by  noting  the 
lengths  marked  off  on  the  chronograph  record  by  a  pen 
point  which  is  given  a  lateral  jerk  electrically  at  the 
beginning  and  end  of  the  interval.  Every  magnitude 
measured  off  on  a  dial  is  finally  referred  to  length,  as 
exemplified  by  pressure  gauges,  gas  meters,  electric  me- 
ters,, aneroid  barometers,  etc. 

Temperature  is  measured  off  as  a  hngth  on  the  stem  of 
the  thermometer. 


ON   MEASUREMENT  3 

Atmospheric  pressure  is  measured,  and  even  expressed, 
in  inches  or  centimeters  of  mercury. 

Weight  is  measured,  in  the  final  adjustment,  by  the 
position  of  a  sHde  or  rider  on  a  linear  scale,  or  in  refined 
work  by  the  position  of  the  balance  pointer  at  equilibrium, 
the  sensibility  of  the  balance  being  known.  The  common 
spring  balance  and  its  more  refined  near  relative,  the  Jolly 
balance,  illustrate  the  linear  principle  in  another  way. 

In  short,  every  measuring  instrument  has  some  sort  of 
linear  scale,  either  straight  or  curved,  on  which  some  sort 
of  indicator  or  pointer  moves. 

The  reason  for  thus  referring  every  kind  of  measurement 
to  a  simple  one  of  length  is  mainly  the  one  already  referred 
to,  that  length  is  the  only  kind  of  magnitude  that  can 
be  conveniently  compared  directly  with  its  own  unit. 
But  there  is  another  reason.  The  eye  can  estimate  a 
length  with  far  greater  accuracy  than  the  muscles  can 
estimate  a  weight,  the  hand  a  temperature,  or  the  con- 
sciousness an  interval  of  time ;  and  this  process  of  esti- 
mation plays  an  all-important  part,  as  will  now  be  seen, 
in  every  kind  of  accurate  measurement. 

3.  Estimation.  —  The  degree  of  precision  with  which 
an  observer  can  read  a  given  linear  scale  depends  upon 
two  things,  namely:  (1)  the  definiteness  or  sharpness  of 
the  marks  on  the  scale  and  of  the  pointer  or  indicator, 
and  (2)  the  skill  with  which  the  observer  can  estimate 
fractional  parts  of  one  interval  or  scale-division. 

The  former  item  may  be  made  clear  by  comparing  the 


4  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

scale  and  indicator  of  an  ordinary  spring  balance  with  those 
on  a  delicate  ampere  meter  or  aneroid  barometer;  or 
the  graduations  on  a  surveyor's  leveling-rod  with  those  on 
a  silver-inlaid  standard  meter  bar. 

As  to  the  second  matter,  it  is  of  the  utmost  importance 
that  the  observer  drill  himself  in  this  process  of  estimation. 
In  no  case  does  the  accuracy  of  a  single  scale-reading  end 
with  the  fineness  of  graduation  of  the  scale,  providing 
the  scale  lines  and  indicator  are  sharp  and  distinct;  it 
can  always  be  carried  a  step  farther. 

It  is  the  custom  of  practical  observers  to  make  estima- 
tions of  fractional  units  in  tenths,  not  in  halves,  thirds, 
etc.,  and  to  record  the  readings  decimally.  No  attempt 
is  made  to  estimate  the  hundredths,  unless  it  appears  to 
the  observer  that  the  fraction  is  exactly  one-fourth  or 
three-fourths,  when  he  would  be  likely  to  record  .25  or 
.75  ;  even  this  is  a  doubtful  practice.  The  reading  of  any 
linear  scale  may  be  carried,  in  general,  to  an  accuracy  of 
one-tenth  of  the  smallest  scale-division  by  the  estimation 
of  the  eye  alone,  or  the  eye  aided  by  a  magnifier  if  desir- 
able. 

In  many  instruments  of  precision,  the  linear  scale  is 
provided  with  some  sort  of  vernier,  which  is  a  mechanical 
substitute  for  the  estimation  of  fractional  parts  of  scale 
divisions.  Descriptions  of  the  different  kinds  of  verniers 
in  use  may  be  found  in  any  elementary  laboratory  manual 
of  physics,  or  in  any  encyclopedia.  But  even  the  use  of 
the  vernier  requires  the  same  sort  of  skill  and  judgment  as 
estimation,  namely,  a  correct  idea  of  linear  position  and 


ON   MEASUREMENT  5 

coincidence.  And  in  the  vast  majority  of  measuring 
instruments,  no  vernier  is  provided,  and  the  observer 
must  be  able  to  estimate  tenths  accurately  and  without 
hesitation. 

4.  The  Impossibility  of  Exact  Measurements.  —  Every 
scientist  is  familiar  with  the  fact  that  there  is  no 
such  thing  as  an  absolutely  exact  measurement,  for  the 
simple  reason  that  the  quantity  measured  and  the  unit 
of  measure  are  never  commensurable. 

If  we  weigh  carefully  a  small  piece  of  metal  on  a  common 
balance,  a  typical  result  would  be  3.9843  grams,  and  not  a 
whole  number,  as  four  grams.  This  is,  however,  only  an 
approximation  to  the  true  weight,  even  if  correct  to  four 
decimal  places,  just  as  the  number  3.1416  is  only  an  apn 
proximation  to  the  value  of  tt.  If  a  more  sensitive  bal- 
ance is  used,  the  result  may  be  3.984326  grams ;  but  as  the 
masses  of  the  piece  of  metal  and  the  gram  weight  are  in- 
commensurable, the  true  weight,  even  if  it  were  possible 
to  weigh  without  the  inaccuracies  that  arise  from  im- 
perfect apparatus  and  judgment  in  estimation,  would  be 
inexpressible  in  grams,  and  the  result  obtained  could 
be  true  only  to  the  degree  of  approximation  represented 
by  six  places  of  decimals,  that  is,  to  the  nearest  millionth 
of  a  gram.       * 

What  is  true  of  weighing  is  true  of  all  measurement,  and 
it  will  readily  be  seen  that  to  obtain  the  true  value  of  any 
actual  concrete  quantity  is  as  hopeless  as  to  obtain  the 
true  value  of  V2^  or  tt,  or  logio  17. 


6      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

5.  Errors  of  Measurement.  —  Aside  from  the  mere 
incommensurability  of  magnitudes,  there  is  another  and 
far  more  serious  hindrance  to  the  obtaining  of  correct 
values  by  measurement,  and  this  is  what  is  technically 
known  as  error. 

Suppose  the  bit  of  metal,  which  was  found  on  the  more 
sensitive  balance  to  weigh  3.984326  grams,  be  now  weighed 
again,  by  the  same  person,  in  the  same  room,  on  the  same 
balance  and  with  the  same  weights.  More  likely  than 
not,  the  result  will  turn  out  to  be  different  from  the  former 
result  by  some  millionths  of  a  gram,  perhaps  thirty  or 
forty  millionths.  This  means  simply  that  neither  re- 
sult is  correct,  even  to  the  sixth  decimal  place. 

Again,  if  we  go  out  with  a  surveyor's  transit  of  the  finest 
construction  and  measure  with  the  utmost  care,  to  seconds 
even,  each  of  the  three  angles  of  a  triangle  marked  out 
by  accurately  centered  stakes  on  level  ground,  and  add 
the  three  results  together,  we  shall  probably  find  that 
their  sum  differs  from  180°  by  several  seconds  one  way  or 
the  other.  We  may  repeat  the  operation  with  equal 
care  and  skill,  and  get  a  still  different  result,  perhaps 
farther  from  180°  than  the  first.  This  illustration  will  be 
all  the  more  striking,  in  that  in  this  case  the  true  value 
of  the  sum  of  the  three  angles  is  known  from  geometry, 
while  in  the  case  of  the  weights  the  true  value  is  not 
and  can  never  be  known.  Even  here,  the  individual 
angles  cannot  be  obtained  exactly. 

The  causes  of  error  in  precise  measurements  are  many 
and  various.     A  single  example  will  suffice  to  illustrate 


ON   MEASUREMENT  7 

this.  Suppose  we  wish  to  measure  the  distance  from  one 
stake  to  another  with  a  surveyor's  chain.  Two  men 
carry  the  chain.  Each  time  they  advance,  one  adjusts  the 
following  end  to  the  rear  marking-pin,  the  other  sets  a 
new  pin  at  the  leading  end,  and  neither  can  do  this  work 
with  absolute  accuracy.  They  do  not  stretch  the  chain 
tight  enough;  they  do  not  hold  the  chain  horizontal  in 
going  up  or  down  hill ;  they  do  not  follow  a  straight  line ; 
they  do  not  notice  kinks  in  the  chain,  and  they  neglect 
the  fact  that  the  chain  is  wearing  at  the  joints  and  getting 
longer.  As  a  consequence  of  all  these  small  items,  and 
many  others  not  mentioned,  the  measurement  may  in 
the  end  be  several  inches  from  the  truth  if  the  line  to  be 
measured  be  very  long.  This  is  only  one  instance  showing 
how  hundreds  of  little  disturbances  may  combine  and 
form  one  final  resultant  error  which  may  be  positive  or 
negative,  great  or  small,  according  to  which  kind  of  dis- 
turbances predominates  (that  is,  whether  they  tend  to 
make  the  result  too  large  or  too  small),  and  to  whether 
they  happen  to  be  about  evenly  balanced  or  not. 

A  systematic  study  of  the  occurrence  of  errors  gives  rise 
to  a  mathematical  analysis,  based  essentially  upon  the 
principles  of  probability  and  known  as  the  Theory  of 
Errors;  and  our  attempts  to  apply  this  theory  to  the  re- 
sults of  measurements,  with  a  view  to  getting  the  values 
that  are  probably  nearest  the  truth,  have  resulted  in 
the  formulation  of  certain  rules  embraced  in  that  part  of 
the  error  theory  known  as  the  Method  of  Least  Squares. 


8      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

EXERCISES 

6.  The  following  exercises  are  intended  for  the  use 
of  students  who  have  not  done  much  laboratory  work  nor 
had  the  advantage  of  a  course  in  laboratory  measurements 
or  field  work.  It  will  be  seen  that  they  are  largely  sug- 
gestive, and  they  may  be  modified  as  desired  to  suit  the 
circumstances.  For  advanced  students  and  research 
workers  they  may  be  omitted  altogether. 

1.  Can  you  think  of  any  kind  of  accurate  measure- 
ment not  ultimately  employing  some  sort  of  linear 
scale  ? 

2.  Show  wherein  the  following  kinds  of  measurements 
are  made  to  employ  a  linear  scale :  area  of  a  piece  of  land ; 
density  of  a  solid ;  relative  humidity  of  the  atmosphere ; 
index  of  refraction  of  a  transparent  substance ;  volume  of 
liquid  from  a  burette. 

3.  Determine  the  volume  of  a  material  sphere,  cylinder 
or  other  geometrical  solid  in  two  ways  :  first  by  measuring 
its  dimensions;  and  second  by  dropping  it  into  a  glass 
graduate  partly  filled  with  water  and  observing  the 
displacement.  Do  the  two  results  agree  ?  Which  do  you 
consider  the  more  precise  method  ? 

4.  Measure  a  quantity  of  pure  water  in  two  ways: 
first  by  placing  it  in  a  glass  graduate;  and  second  by 
weighing  it  on  a  balance  and  computing  the  volume. 
The  weighing  may  be  done  in  the  graduate,  which  has 
been  weighed  beforehand. 


ON    MEASUREMENT  9 

5.  Lay  off  on  a  sheet  of  smooth  paper,  with  a  fine,  hard 
pencil,  a  Hne  of  indefinite  length,  and  mark  two  points  on 
it  at  random  somewhat  less  than  10  cm.  apart.  On  the 
straight  edge  of  a  card,  mark  two  points  as  nearly  10  cm. 
apart  as  possible.  By  means  of  direct  comparison  with 
this  standard,  estimate  the  length  of  the  first  line-segment 
in  centimeters,  writing  down  the  result.  Next  compare 
the  unknown  line  with  a  cardboard  scale  marked  off  in 
centimeters  but  not  in  millimeters,  observing  the  num- 
ber of  centimeters  and  estimating  the  millimeters.  Finally 
compare  the  same  line  with  a  millimeter  scale,  estimating 
the  tenths  of  a  millimeter.  Notice  how  the  three  results 
agree,  all  being  expressed  in  centimeters.  Repeat  this 
several  times  with  different  line-segments. 

6.  Devise  and  perform  exercises,  similar  to  Exer- 
cise 5,  in  the  measurement  of  angles,  using  a  large 
protractor  and  circular  sectors  of  paper  as  measuring  in- 
struments. 

7.  Try  measuring  short  intervals  of  time  to  tenths  of 
a  second  by  means  of  an  ordinary  watch.  In  order  to  test 
the  results,  let  the  period  measured  be  the  time  of  swing 
of  a  simple  pendulum,  and  measure  by  the  watch  intervals 
of  five,  ten,  fifteen,  twenty  and  one  hundred  swings,  find- 
ing the  time  of  a  single  vibration  from  each  measurement. 
Do  the  results  agree?  Have  you  any  greater  confidence 
in  one  than  in  another  ? 

8.  Familiarize  yourself  thoroughly  with  the  use  of  as 
many  different  kinds  of  verniers  as  are  available.     Before 


10     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

using  the  vernier  in  each  case,  estimate  the  fraction  of  a 
unit  in  tenths  by  the  eye. 

9.  Weigh  a  small  piece  of  iron  by  means  of  a  Jolly  bal- 
ance, then  on  a  trip  scale,  then  on  an  equal  arm  balance. 
Compare  the  results.  In  which  result  have  you  the  great- 
est confidence?     Why? 

10.  Weigh  a  small  object  several  times,  with  the  high- 
est degree  of  precision  attainable,  on  a  good  balance.  The 
pointer  method  should  be  used.     Are  the  results  all  equal  ? 


CHAPTER  II 

ON  THE   OCCURRENCE  AND   GENERAL 
PROPERTIES    OF   ERRORS 

7.  Errors  and  Residuals.  —  The  term  error  has  so  far 
been  used  somewhat  indefinitely,  and  it  will  be  necessary, 
before  going  further,  to  explain  its  exact  meaning,  as  well 
as  that  of  another  term  closely  connected  with  it. 

We  have  seen  that  different  measurements  upon  the 
same  quantity  generally  give  different  results.  These 
results  evidently  cannot  all  be  correct,  and  it  is  very  un- 
likely that  any  of  them  is  correct,  even  to  the  degree  of 
precision  (that  is,  to  the  number  of  decimal  places)  attain- 
able with  the  instruments  and  method  used.  The  differ- 
ence between  the  result  of  an  observation  and  the  true 
value  of  the  quantity  measured  is  called  the  error  of  the 
observation.  In  what  follows  we  shall  generally  denote 
observations  by  the  symbol  s,  the  quantities  upon  which 
they  are  made  by  q,  and  the  errors  of  the  observations  by 
X,  the  latter  being  defined,  as  just  stated,  as  the  difference 

X  =  s  -  q,  (1) 

which  will  be  positive  or  negative  according  as  the  observa- 
tion is  too  large  or  too  small.  The  student  should  be  care- 
ful to  remember  this  definition,  and  to  apply  it  to  such  illus- 

11 


12     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

trations  as  the  following  :  If  a  line  is  exactly  437  feet  long, 
and  the  result  of  a  measurement  upon  it  is  436.2  feet,  then 
the  error  is  436.2  -  437  =  -  0.8  ft. 

While  we  cannot  ordinarily  obtain  the  true  value  of  a 
measured  quantity  from  one  measurement,  nor  even  by 
averaging  many  measurements,  the  method  of  least  squares 
furnishes  us,  in  the  latter  case,  with  a  means  of  calculating 
what  is  called  the  most  probable  value,  which  is  the  closest 
approximation  to  the  true  value  that  the  series  of  obser- 
vations is  capable  of  yielding.  A  familiar  illustration  is 
that  of  a  series  of  direct  observations  upon  a  single  quan- 
tity, in  which  case  the  most  probable  value  is  simply  the 
arithmetical  average  of  the  several  results. 

Having  obtained  the  most  probable  value  from  a  series 
of  observations  in  the  manner  hereafter  to  be  explained, 
if  we  now  subtract  it  from  each  measured  result,  we  obtain 
a  series  of  differences  known  as  the  residuals  corresponding 
to  the  respective  observations.  The  most  probable  value 
being  denoted  by  m,  and  any  observation  by  s,  the  resid- 
ual corresponding  to  s  is 

p  =  s  —  m.  (2) 

Thus,  the  residual  bears  the  same  relation  to  the  error  that 
the  most  probable  value  bears  to  the  true  value.  If  the  num- 
ber of  observations  be  very  large,  and  the  observations  be 
very  precise,  then  the  most  probable  value  may  be  very, 
very  close  to,  though  never  equal  to,  the  true  value;  and 
in  that  case  the  residuals  will  be  equally  close  to  the  cor- 
responding true  errors. 


PROPERTIES   OF  ERRORS  13 

It  is  worthy  of  note  that,  since  the  true  value  of  a  quan- 
tity in  terms  of  any  arbitrarily  selected  unit  is  always  an 
incommensurable  number,  while  the  most  probable  value 
is  commensurable,  it  follows  that  the  error  of  any  observa- 
tion is  incommensurable,  while  the  corresponding  residual 
is  commensurable.  The  true  value  and  the  errors  are  con- 
sequently forever  unknown  and  figure  only  in  theoretical 
discussions  (with  such  exceptions  as  have  been  noted) ; 
and  we  deal  in  practice  only  with  their  close  approxi- 
mations, the  most  probable  value  and  the  residuals. 

8.  Classification  of  Errors.  —  It  is  now  very  important 
to  point  out  that  errors  of  observation  may  be  divided 
naturally  into  two  distinct  classes,  whose  occurrence,  and 
the  methods  of  dealing  with  which,  are  entirely  different. 

First  we  may  consider  those  errors  which  arise  from 
causes  that  continue  to  operate  in  the  same  manner 
throughout  the  series  of  observations,  and  which  may 
therefore  be  called  persistent  or  systematic  errors.  In 
many  cases,  persistent  errors  not  only  occur  in  the  same 
manner,  but  have  the  same  value,  throughout  the  investi- 
gation, and  they  may  then  be  called  constant  errors. 

The  causes  of  persistent  errors,  which  are  often  known 
to  the  observer  and  may  in  many  cases  be  eliminated  or 
avoided  by  methods  presently  to  be  explained,  may  be, 
for  the  most  part,  looked  for  under  one  or  another  of  the 
following  heads. 

a.  Incorrect  Instruments.  —  The  instruments  or  scales 
used  may  not  be  true.     For  example,  if  a  100-foot  tape 


14     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

is  actually  only  99.99  feet  in  length,  every  measurement 
on  a  line  made  with  that  tape  will  tend  to  give  a  result  one 
ten-thousandth  too  long,  no  matter  how  many  times  the 
observation  is  repeated;  or  if  a  clock  used  in  scientific 
work  gains  one  second  a  day,  every  measurement  on  an 
interval  of  time  made  w^ith  that  clock  is  just  the  correspond- 
ing fraction  too  long.  (In  each  of  these  cases,  is  the  error 
positive  or  negative  ?) 

b.  Imperfect  Setting  of  Scale.  —  Owing  to  carelessness 
or  accident,  the  scale  on  a  measuring  instrument,  though 
truly  graduated,  may  be  displaced  from  its  proper 
position  by  a  small  amount.  This  is  well  illustrated  by 
the  mercurial  barometer,  on  which  the  scale  must  be 
adjusted  at  each  reading,  to  allow  for  the  rise  and  fall  of 
mercury  in  the  reservoir;  and  a  clock  Which,  though 
running  at  the*  proper  rate,  has  been  set  a  little  ahead  or 
behind  the  true  standard  time,  is  an  analogous  case. 

c.  Defective  Mechanism.  —  No  instrument  is  absolutely 
perfect  from  a  mechanical  standpoint,  and  every  instru- 
ment of  precision  must  be  frequently  tested  if  we  would 
rely  upon  the  results  of  its  use.  The  arms  of  a  balance  are 
never  really  equal,  and,  what  is  worse,  they  are  continually 
changing  their  relative  length,  owing  to  changes  of  tem- 
perature. Nor  has  it  been  found  possible  to  construct 
a  clock  that  will  run  with  absolutely  constant  rate,  even 
at  a  constant  temperature  and  in  a  vacuum. 

d.  Fabe  Indicator  Settings.  —  In  very  delicate  instru- 
ments, such  as  the  balance  or  the  aneroid  barometer, 
the  indicator  frequently  comes  to  rest,  on  account    of 


PROPERTIES   OF   ERRORS  15 

friction,  in  a  false  position.  In  the  case  of  the  aneroid 
barometer,  it  often  suffices  to  tap  the  dial  gently,  in  order 
to  make  the  indicator  assume  its  true  position.  The  same 
may  be  said  of  the  magnetic  compass-needle. 

e.  Knoivn  External  Disturbances.  —  It  is  often  the  case 
that  persistent  errors  are  introduced  by  external  causes 
whose  nature  is  well  understood,  but  which  cannot  be 
avoided.  Thus,  a  heated  body  under  experimental  in- 
vestigation always  radiates  some  heat,  in  spite  of  the  most 
elaborate  precautions ;  and  the  length  of  a  measuring  rod 
or  tape  is  certain  to  vary  with  changes  of  temperature. 

f .  Personal  Equation  and  Prejudice.  —  Every  observer 
exhibits  peculiarities  or  habits  of  observation  which  cause 
him  to  have  a  tendency  toward  persistent  error  in  the 
same  direction.  Thus,  one  observer  may  continually 
overestimate  in  the  estimation  of  tenths,  another  will  under- 
estimate ;  a  time  observer  requires  a  certain  definite 
interval  to  respond  to  a  stimulus,  that  is,  to  obey  a  signal 
of  any  sort.  This  unconscious,  persistent  error  on  the 
part  of  an  observer  is  called  his  personal  equation. 

Somewhat  analogous  to  personal  equation  is  what  may 
be  called  prejudice.  After  an  observer  has  made  one 
measurement  of  a  quantity  on  a  fixed  scale,  and  made  the 
estimation  of  tenths,  there  is  a  natural  tendency  for  him 
to  allow  his  first  estimation  to  affect  the  subsequent  ones. 
This  difficulty  is  often  met  with  in  the  use  of  the  vernier, 
where  it  is  necessary  to  judge  as  to  which  line  coincides 
most  nearly  with  its  fellow  on  the  scale^ 

The  second  class  of  errors  referred  to  at  the  beginning 


16     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

of  this  article  comprises  those  whose  causes  are  temporary, 
existing  through  only  one  observation,  and  disappearing 
entirely  upon  a  slight  change  of  conditions.  Such  errors 
are  not  recognizable,  and  sometimes  not  even  suspected, 
until  their  existence  is  demonstrated  by  the  discrepancies 
between  successive  observations  when  all  known  disturb- 
ances have  been  eliminated.  These  are  known  as  acci- 
dental errors. 

Accidental  errors  may  also  be  subdivided,  as  follows. 

a.  Those  Due  to  External  Causes.  —  Accidental  errors 
may  result  from  causes  entirely  foreign  to  the  observer 
and  of  so  complex  a  character  as  to  be  incapable  of  analysis. 
For  example,  in  sighting  a  mark  with  a  surveyor's  transit, 
a  sudden  gust  of  wind  may  imperceptibly  sway  the  in- 
strument for  a  moment,  or  someone  may,  without  the  ob- 
server's knowledge,  knock  against  the  tripod  and  jar  the 
telescope  slightly  out  of  place.  In  making  delicate  mag- 
netic measurements,  such  rapid  changes  as  often  take 
place  unexpectedly  in  the  earth's  magnetic  field  may 
momentarily  affect  the  equilibrium  of  the  needle.  In 
sighting  at  a  star  with  a  telescope,  currents  of  air  in  the 
upper  atmosphere  may  cause  it  to  waver  and  appear  for 
a  moment  to  one  side  of  its  mean  apparent  position.  In 
using  a  balance,  the  zero  of  equilibrium  may  change 
slightly  during  the  course  of  a  single  weighing,  owing, 
perhaps,  to  an  unsuspected  fluctuation  of  tempera- 
ture. 

It  will  thus  be  seen  that  observations  of  all  kinds  are 
affected  by  multitudes  of  such  causes,  which  are  of  greater 


PROPERTIES  OF  ERRORS  17 

or  less  importance,  but  which  all  tend  to  affect  the  accuracy 
of  the  results. 

b.  Accidental  Errors  of  Judgment.  —  Aside  from  per- 
sonal equation  and  prejudice,  the  observer  himself  is  sub- 
ject to  fluctuations  of  judgment,  both  as  to  the  adjustment 
of  his  instrument  and  as  to  the  estimation  of  tenths.  An 
attempt  to  analyze  in  detail  the  causes  of  these  internal 
tendencies  to  err  in  judgment  would  belong  to  the  realm 
of  psychology ;  but  we  may  mention  as  prominent  among 
them  the  influences  of  imperfect  vision,  optical  illusion, 
inattention  and  fatigue,  the  last  mentioned  cause  probably 
affecting  the  others  in  a  very  large  degree. 

Some  of  the  methods  commonly  employed  in  dealing 
with  persistent  errors  are  briefly  mentioned  in  Art.  10. 
It  is,  however,  the  study  of  accidental  errors,  and  of  the 
laws  which  are  found  to  govern  their  occurrence,  that 
constitutes  the  special  office  of  the  method  of  least  squares. 

9.  Mistakes.  —  Entirely  distinct  from  errors,  in  the 
sense  heretofore  used,  are  those  inaccuracies  which  are 
due  purely  to  carelessness,  and  which  should  properly  be 
called  mistakes.  They  consist  in  such  blunders  as  reading 
the  wrong  number  on  the  scale,  reading  one  number  and 
putting  another  down  in  the  notes,  reading  a  vernier  back- 
ward instead  of  forward,  making  a  miscount  in  timing 
a  pendulum,  etc.  Mistakes  are  usually  easily  detected, 
and  there  is  no  remedy  except  vigilance  and  careful  check- 
ing. When  measurements  are  made  more  than  once  the 
checking  is  a  simple  matter. 


18  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

10.  General  Methods  of  Eliminating  Persistent  Errors. 
—  In  Art.  8  are  enumerated  several  causes  of  persistent 
errors,  with  illustrations  of  each.  Though  their  discussion 
does  not  properly  belong  to  the  general  theory  of  errors, 
it  may  not  be  out  of  place  to  describe  here  some  of  the 
methods  commonly  employed  in  dealing  with  them, 
especially  as  the  theory  of  errors  is  frequently  applied  in 
the  processes  of  correction  here  referred  to.  The  treat- 
ment of  the  several  sources  of  persistent  errors  will  be 
taken  up  in  the  same  order  as  they  are  mentioned  in 
Art.  8,  and  designated  by  the  same  letters. 

a.  Incorrect  Instruments.  Adjustment  and  Standardiza- 
tion. —  As  it  is  never  certain  that  an  instrument  measures 
in  true  units,  it  is  necessary  to  test  it  before  relying  upon 
the  results  of  its  use.  (The  tests  may  in  some  cases  be 
made  long  after  the  measurements.)  An  instrument  may 
sometimes  be  adjusted  correctly,  and  remain  so;  more 
commonly  it  gets  out  of  adjustment  again,  from  wear  or 
other  causes.  Actual  adjustment  may  often  be  inconvenient 
or  impossible.  A  more  approved  practice  is  standardizor 
Hon,  which  will  apply  to  nearly  every  case.  This  consists  in 
comparing  the  instrument  with  a  standard  and  determining 
the  true  value  of  each  of  its  scale  divisions  or  units,  and 
then,  instead  of  trying  to  adjust  the  instrument,  simply 
making  the  necessary  corrections  on  the  observations. 
(Where  standardization  extends  over  a  whole  scale,  it 
is  commonly  called  calibration.)  Thus,  the  astronomer 
seldom  corrects  his  clock ;  he  simply  determines  its  error 
from  the  stars  at  intervals,  and  thus  deduces  its  error  in 


PROPERTIES  OF  ERRORS  19 

rate,  which  is  all  the  information  needed  at  any  time. 
Laboratory  weights  are  seldom  correct  when  purchased, 
and  moreover  they  lose  or  gain  weight  by  wear  or  corro- 
sion ;  hence  they  should  be  compared  from  time  to  time 
with  standards  kept  for  the  purpose.  Numerous  illustra- 
tions of  the  kind  will  occur  to  the  reader. 

b.  Imperfect  Setting  of  Scale.  Differential  Method.  — 
The  error  due  to  imperfect  setting  of  the  scale  may  often 
be  eliminated  by  the  differential  method,  which  consists 
in  reading  the  position  of  the  indicator  when  it  should  be 
at  zero,  then  again  when  it  is  affected  by  the  quantity  to 
be  measured,  and  taking  the  difference.  This  method 
applies  only  when  the  scale  divisions  are  equal  throughout 
the  scale.  The  process  is  one  very  generally  employed, 
as  it  has  further  advantages  than  the  one  here  stated; 
very  frequently  it  is  the  only  method  practicable.  The  use 
of  a  level  and  leveling  rod  in  surveying  illustrates  the  latter 
point,  as  does  almost  any  kind  of  comparator ;  and  when 
one  wishes  to  weigh  a  portion  of  liquid,  he  must  needs 
subtract  the  weight  of  the  empty  vessel  from  the  weight 
of  the  vessel  and  contained  liquid. 

c.  Defective  Mechanism.  Compensation.  —  Instru- 
mental errors  may  often  be  made  to  react  against  them- 
selves and  automatically  disappear.  When  this  can  be 
done,  it  is  by  far  the  best  method  of  elimination.  A 
simple  example  is  the  process  of  "  double  weighing,"  in 
which  the  effect  of  inequality  in  the  arms  of  the  balance 
is  removed  by  weighing  with  the  object  first  on  one  pan, 
then  on  the  other,  and  taking  the  mean.     (Strictly,  the 


20     THEORY    OF   ERRORS   AND    LEAST   SQUARES 

geometrical  mean  should  be  used.)  If  a  spirit  level, 
resting  upon  an  imperfectly  adjusted  base,  be  simply 
reversed,  end  to  end,  the  half-way  point  between  the  two 
positions  of  the  bubble  will  indicate  its  true  position  as 
well  as  if  it  were  in  adjustment.  The  graduated  circles 
used  on  surveying  instruments,  spectrometers,  and  the 
like,  are  usually  provided  with  two  diametrically  opposite 
verniers,  so  that  the  error  arising  from  the  vernier  system 
being  out  of  center  with  the  circle  itself  may  disappear 
on  taking  the  mean  of  the  readings  of  the  two  verniers. 
In  using  a  galvanometer  it  is  well  to  reverse  the  current 
and  read  the  deflection  both  ways  on  the  scale.  An  in- 
teresting application  of  the  method  to  the  elimination  of 
unknown  external  disturbance  is  the  scheme  devised  by 
Rumford  for  neutralizing  the  effect  of  radiation  in  calori- 
metric  measurements.  A  preliminary  experiment  is  made 
to  determine  by  what  amount  the  temperature  of  the  calo- 
rimeter will  be  raised ;  and  then  the  initial  temperature  is 
so  adjusted  that  it  is  about  the  same  amount  beloiv  the 
temperature  of  the  surrounding  air  at  the  beginning  of 
the  experiment  as  it  is  above  it  at  the  close,  so  that  practi- 
cally the  same  amount  of  heat  is  absorbed  during  the  first 
half  of  the  operation  as  is  radiated  during  the  last  half. 

d.  False  Indicator  Settings.  Oscillation.  —  In  cases 
where  the  indicator  comes  to  rest  in  a  false  position,  due 
to  friction,  the  difficulty  may  often  be  removed  by  not 
allowing  the  indicator  to  come  to  rest  at  all,  but  reading 
it  while  still  oscillating.  This  method  has  the  further 
advantage  of  saving  time  in  such  instruments  as  the  bal- 


PROPERTIES  OF  ERRORS  21 

ance  and  undamped  galvanometers  or  magnetometers. 
In  order  to  compensate  for  diminishing  amplitude,  one 
more  reading  should  be  taken  at  one  extreme  of  the  swing 
than  at  the  other,  as  in  the  following  balance  pointer 
readings  and  reduction: 


Left 

Right 

7.8 

13.1 

8.0 

13.0 

8.1 

2)26.1 

3)23.9 

13.05 

7.97 

13.05 

2)21.02 

10.51 

True  reading. 

This  result  is  much  more  quickly  obtained  and  more 
accurate  than  one  obtained  by  letting  the  pointer  come  to 
rest. 

e.  Theoretical  Corrections  for  Known  External  Disturb- 
ances. —  When  the  manner  in  which  external  disturbances 
operate  is  known,  and  their  magnitude  determined,  the 
errors  due  to  them  are  eliminated  by  simply  applying  the 
computed  corrections.  The  temperature  and  stretch 
corrections  applied  to  the  steel  tape  in  precise  chaining, 
and  the  temperature  corrections  necessary  with  instru- 
ments, such  as  the  barometer  and  pyknometer,  depending 
upon  the  density  of  a  liquid  or  the  capacity  of  a  hollow 
vessel,  are  familiar  examples.  Instead  of  employing 
Rumford's  compensation  in  using  the  calorimeter,  the 
amount  of  radiation  per  minute  may  be  previously  noted 


22     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

and  allowed  for  in  the  reduction  of  the  results.  The  re- 
fraction error  in  the  observed  altitude  of  a  star,  or  in  long 
range  leveling,  the  vacuum  correction  in  weighing,  etc., 
are  further  familiar  examples.  It  is  for  the  purpose  of 
obtaining  data  for  such  corrections  that  many  investiga- 
tions of  the  behavior  of  physical  phenomena  under  varying 
conditions  are  carried  on ;  indeed,  this  work  constitutes  a 
large  part  of  quantitative  scientific  research. 

f .  Corrections  for  Personal  Equation  and  Prejudice.  — 
Personal  equation  may  be  eliminated,  either  by  deter- 
mining by  means  of  specially  devised  experiments  what 
the  personal  equation  of  the  observer  is  for  a  given  kind  of 
measurement,  or  by  arranging  matters  so  as  to  make  the 
personal  error  act  in  opposite  directions  in  the  two  halves 
of  the  observation ;  or  by  a  very  different  method,  — 
that  of  employing  a  number  of  different  observers  on  the 
same  measurement,  whose  errors  will  tend  to  compensate 
in  the  long  run,  like  accidental  errors. 

The  effect  of  prejudice  may  often  be  avoided  by  altering 
the  conditions.  Thus,  when  repeatedly  using  the  differen- 
tial method,  the  whole  measurement  may  be  shifted  each 
time  to  a  different  part  of  the  scale.  The  oscillation  method 
is  not  subject  to  prejudice,  since,  though  the  true  reading 
may  be  the  same  in  the  successive  observations,  the  oscilla- 
tions approaching  it  will  not  be.  An  experienced  observer 
will  not  allow  prejudice  to  influence  him  to  any  great  extent. 

11.  Exercises  Leading  to  an  Understanding  of  Error 
Distribution.  —  Before  attempting  any  introduction  to  the 


PROPERTIES  OF  ERRORS 


23 


methods  of  dealing  with  accidental  errors  in  measurement, 
it  is  necessary  that  the  student  recognize  the  existence  of 
a  law  governing  their  occurrence,  and  become  to  some 
extent  familiar,  through  experience,  with  the  operations 
of  that  law.  To  this  end,  it  is  deemed  worth  while  to 
introduce  at  this  point  a  number  of  laboratory  exercises 
or  experiments,  in  which  the  phenomenon  to  be  studied  is 
the  distribution  of  errors  as  governed  by  the  law  of  chance. 
The  term  "  laboratory  "  refers  to  the  method  only ;  the 
exercises  may  be  performed  at  one's  study  table  without 
any  special  apparatus. 

1.  No  better  analogy  to  the  behavior  of  accidental 
errors  can  be  found  than  in  the  manner  in  which  shots 
fired  at  a  target  are  found  to  distribute  themselves  with 
respect  to  a  point  fired  at.  To  illustrate  this  experi- 
mentally, take  a  sheet  of  ordinary  foolscap  or  other  ruled 
paper  and  with  a  black  pencil  make  the  ruled  line  nearest 
the  middle  of  the 
sheet  heavier  than 
the  others,  so  as  to 
be  distinctly  visible 
a  few  feet  away. 
Lay  the  paper  on  a 
board  or  smooth 
book,  and  place  it, 
face  upward,  on  the 
floor.  Take  a  rather  long  pencil  lightly  between  the  ex- 
tended finger-tips  of  both  hands,  and  standing  with  the 
eye  directly  over  the  black  line  on  the  paper,  hold  the 


Fig.  1 


24     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

pencil,  point  downward,  over  the  line,  and  endeavor  to 
drop  it  so  as  to  strike  the  line  with  the  descending  pencil- 
point.  In  other  words,  make  the  central  line  a  target; 
the  shots  will  be  self-recorded  by  the  dots  on  the  paper. 
Take  at  least  a  hundred  shots  in  this  manner,  each  time 
trying  with  all  possible  skill  to  hit  the  central  line.  Having 
done  this,  prepare  another  sheet  of  paper  ruled  off  in  a 
similar  manner  (ordinary  coordinate  paper  will  do)  and 
plot  on  it  a  curve  whose  ordinates  represent  the  relative 
number  of  shots  found  to  have  struck  in  each  compart- 
ment of  the  ruled  target  and  whose  abscissas  represent  the 
distances  of  the  respective  compartments  from  the  central 
line.  In  case  a  shot  appears  to  have  struck  exactly  upon 
one  of  the  lines,  assign  it  to  the  compartment  on  the  side 
toward  the  center. 

Can  you  think  of  any  influence  that  might,  in  this  ex- 
periment, be  analogous  to  a  persistent  error  in  measure- 
ment? What  effect  would  it  have  on  the  curve?  Keep 
the  data  for  future  use. 

2.  On  a  sheet  of  smooth  paper,  draw  a  line  with  a  hard, 
sharp  pointed  pencil  and  mark  two  points  on  it  about  a 
foot  apart.  The  exercise  is  to  measure  this  line  with  a 
metric  scale  to  hundredths  of  a  centimeter,  estimating  the 
hundredths  as  tenths  of  a  millimeter.  In  order  to  avoid 
prejudice,  it  will  be  well  to  place  a  third  point  somewhere 
between  the  others,  and  measure  the  line  in  two  segments, 
a  and  b.  Now  measure  a  and  6  alternately,  using  the 
differential  method,  until  each  has  been  measured,  say,  a 
hundred  times.     Add  the  corresponding  pairs  of  values 


PROPERTIES  OF  ERRORS  25 

and  record  the  sums  as  the  measured  lengths  of  the  Une. 
Find  the  mean  of  the  hundred  values  to  the  nearest  hun- 
dredth of  a  centimeter,  and  record  the  departure  from  it 
of  each  of  the  observations,  plus  or  minus.  These  de- 
partures are  the  residuals  of  the  observations  (Art.  7).  It 
will  be  noticed  that  a  large  number  of  residuals  have  the 
same  value.  Determine  how  many  there  are  of  each 
value,  separating  positive  from  negative,  and  plot  a  curve 
whose  abscissas  represent  the  values  of  the  residuals  and 
whose  ordinates  represent  the  numbers  of  residuals  having 
those  respective  values.  A  convenient  scale  should  be 
used :  for  example,  on  the  abscissas,  let  1  cm.  represent 
0.1  mm.  of  residual,  and  on  the  ordinates,  let  each  residual 
be  represented  by  a  millimeter.  Keey  the  data  for  future 
iLse. 

What  change  would  have  to  be  made  in  the  curve  if 
the  abscissas  and  ordinates  were  the  values  and  numbers, 
respectively,  of  the  true  errors  instead  of  the  residuals, 
supposing  that  there  is  any  means  of  knowing  the  former  ? 

3.  The  preceding  exercise  may  be  varied  by  using,  for 
the  measured  quantity,  an  angle  of  exactly  180°,  measuring 
it  in  two  segments  with  a  protractor  to  tenths  of  a  degree. 
In  this  case  the  true  value,  and  hence  the  true  errors,  are 
known.     Keep  the  data. 

4.  Do  the  curves  obtained  from  the  preceding  exercises 
bear  any  resemblance  to  each  other  ?  Construct  a  smooth 
curve  which  seems  to  be  typical  of  them.  Does  this  curve 
resemble  any  familiar  geometrical  form?  Plot  the  curve 
y  =  2"^,  taking  10  cm.  as  the  unit    for  both  abscissas 


26     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

and  ordinates  and  assigning  to  x  the  successive  values  0, 
0.1,  0.2,  0.3,  etc.,  both  positive  and  negative. 

5.  From  the  results  of  the  foregoing  exercises,  does 
there  appear  to  be  any  relation  connecting  the  magnitude 
pf  an  error  with  the  frequency  of  its  occurrence?  Can 
you  assign  any  reason  for  such  a  relation?  Do  positive 
errors  appear  to  occur  any  more  frequently,  in  the  long 
run,  than  negative  errors,  or  vice  versa  f 


12.  Remarks  on  the  Distribution  of  Errors. — The  curve 
to  which  the  preceding  exercises  have  introduced  us  is 
commonly  called  the  probability  curve,  though  a  better 

name  would  be  the 
curve  of  departures, 
as  will  appear  later. 
Superficially  it  some- 
what resembles  the 
"  witch,"  a  typical 
case  being  shown  in 
Fig.  2.  The  student 
must  not  expect  that 
any  curve  plotted  from  the  results  of  such  experiments 
as  the  foregoing  will  be  smooth  and  regular,  like  the  curve 
here  shown;  actual  curves  are  broken  and  irregular. 
But  the  greater  the  number  of  observations  or  data,  the 
nearer  will  the  actual  departure  curve  assume  the  smooth, 
symmetrical  form  assigned  by  theory. 

The  results  of  experiments,  as  we  have  seen,  and  theoreti- 
cal considerations,  as  will  appear,  both  point  to  the  follow- 


FiG.  2 


PROPERTIES   OF   ERRORS  27 

ing  facts  regarding  the  distribution  of  accidental  errors, 
all  of  which  may  be  deduced  from  an  examination  of  the 
curve. 

1.  The  frequency  with  which  an  accidental  error  of  given 
magnitude  occurs  depends  upon  the  magnitude  of  the  error. 

2.  Large  errors  occur  less  frequently  than  small  ones. 

3.  The  error  distribution  is  symmetrical;  that  is,  positive 
and  negative  errors  of  the  same  magnitude  occur  with  the 
same  frequency. 

Though  these  laws  do  not  of  course  apply  absolutely 
in  any  one  case,  yet  they  express  the  general  tendency  of 
error,  and,  in  fact,  the  general  tendency  of  all  accidental 
departures  from  the  normal  or  mean,  as,  for  example, 
the  statures  of  individual  people  as  compared  with  the 
average  stature  of  the  race.  In  theoretical  discussions, 
the  number  of  observations  made,  or  of  data  considered, 
is  regarded  as  infinite,  and  the  curve  as  strictly  sym- 
metrical. 

In  the  case  of  measurements,  with  which  we  are  here 
concerned,  if  the  results  are  affected  by  persistent  error 
from  any  source,  they  will  be  found  to  cluster  about  the 
theoretical  most  probable  value  of  the  measured  quantity 
instead  of  the  true  value,  there  being  now  an  appreciable 
difference  between  the  two.  The  whole  curve  of  errors 
now  becomes  a  curve  of  residuals,  and  is  merely  shifted 
a  little  to  one  side  or  the  other  according  as  the  persistent 
error  is  positive  or  negative.  If,  for  example,  in  the  second 
exercise  of  Art.  11,  the  scale  used  had  its  spaces  slightly  too 
long,  the  whole  curve  would  be  shifted  a  little  in  the 


28     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

negative  direction,  simply  because  each  observation  tends 
to  undervalue  the  line  measured  on  account  of  the  defect 
in  the  scale. 

From  this  consideration  it  is  clear  that  when,  as  is  really 
always  the  case,  the  true  value  of  the  measured  quantity 

is  not  given  by 
the  measurements, 
a  study  of  the 
curve  of  residuals 
will  reveal  nothing 
as  to  the  presence 
or  absence  of  per- 
sistent errors.  The 
law  of  probability 
of  error  is  con- 
cerned only  with  accidental  errors,  that  is,  those  whose 
causes  are  of  temporary  duration,  —  the  result,  as  we 
say,  of  pure  chance. 


13.  Precision.  —  On  comparison  of  the  results  of  differ- 
ent sets  of  measurements,  even  upon  the  same  quantity, 
it  is  found  that  the  error  curve  is  not  of  constant  form. 
Every  gradation  is  met  with  (Fig.  4),  from  low,  flat  curves 
to  high,  pointed  ones.  This  peculiarity  may  be  observed 
when  we  make  several  series  of  measurements  upon  the 
same  quantity  by  different  methods.  The  variation  is 
easily  interpreted. 

Compare,  for  example,  curves  A  and  D.  In  the  case  of 
A  there  are  nearly  as  many  large  errors  as  small  ones. 


PROPERTIES   OF   ERRORS 


29 


For  shots  fired  at  a  target,  this  would  indicate  poor  marks- 
manship or  long  range ;  in  measurement,  it  means  random 
judgment,  crude  instruments,  or  circumstances  which 
render  the  work  difficult.  In  the  case  of  D,  on  the  other 
hand,  the  number  of  large  errors  is  very  small,  the  great 
body  of  results  being  crowded  closely  about  the  mean 
and  indicating  its 
position  with  con- 
siderable definite- 
ness.  From  this  it 
is  clear  that  the  form 
of  the  error  or  resid- 
ual curve  depends 
upon  the  precision 
with  which  the  ob- 
servations have  been 
made. 

To  illustrate  what 
is  meant  by  preci- 
sion, let  two  parties 
of  observers  each  make  a  set  of  measurements  on  tlie  dis- 
tance between  two  stakes,  the  one  with  a  ten-foot  pole,  the 
other  with  a  steel  tape.  The  most  probable  value  deduced 
from  one  set  may  not  differ  much  from  that  deduced 
from  the  other,  but  the  residual  curves  plotted  from  the 
two  sets  of  results  will  show  considerable  difference  of 
precision,  mainly  on  account  of  the  larger  number  of 
times  that  the  ten-foot  pole  must  be  laid  down  and  its 
consequent  greater  liability  to  error. 


Fig.  4 


30     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

The  form  of  the  residual  curve  may  therefore  be  used  as 
a  test  of  the  efficiency  of  an  observer,  an  instrument  or 
a  method  of  measurement.  It  will  be  seen  later  that  the 
same  test  can  be  applied  by  means  of  mathematical 
formulas,  without  the  labor  of  plotting  the  curve  (Chapter 
VIII). 

14.   Mathematical  Expression  of  the  Law  of  Error.  — 

The  evident  existence  of  some  law  governing  the  distribu- 
tion of  errors  leads  us  to  inquire  what  that  law  is,  and 
whether  it  is  expressible  by  a  simple  mathematical  rela- 
tion. Some  of  the  facts  concerning  the  behavior  of  errors 
have  already  been  deduced ;  but  the  theoretical  expres- 
sion of  the  law  itself,  and  even  the  very  language  in  which' 
it  is  expressed,  must  be  reserved  until  the  student  has 
reviewed  some  of  the  fundamental  principles  of  the 
theory  of  probabilities  and  has  been  introduced  to  some  of 
the  special  problems  in  probability  upon  which  the  theory 
of  errors  is  found  to  depend.  The  following  chapter  is, 
therefore,  devoted  to  this  subject. 


CHAPTER  III 
ON   PROBABILITIES 

15.  Fundamental  Principle.  —  It  is  a  common  remark 
that  one  thing  is  more  Hkely  to  happen  than  another. 
In  speaking  thus,  one  concedes  that  either  of  the  two 
events  may  happen,  and  attempts  no  prediction  as  to 
which  will  happen,  if  either ;  yet  he  recognizes  a  preponder- 
ance of  the  Hkelihood  of  one  event  over  that  of  the  other. 

In  the  kind  of  magnitude  here  recognized,  that  is, 
likelihood  or  probability,  there  is,  in  the  great  majority 
of  cases,  no  means  of  measuring  or  giving  numerical  ex- 
pression to  its  relative  degrees.  It  is  said  that  corn 
growing  on  low  ground  is  more  likely  to  be  caught  by  frost 
than  that  on  high  ground,  but  there  is  no  means  of  telling 
how  many  times  more  likely  it  is. 

It  is  possible,  however,  to  give  such  a  definite  meaning 
to  the  term  probability  that  the  relative  probabilities  of 
some  simpler  events  may  be  calculated  and  expressed.  In 
framing  such  a  definition,  it  is  necessary  to  recognize  an 
important  principle  in  the  operation  of  chance,  governing 
the  behavior  of  events  whose  causes  are  at  least  partly 
manifest,  and  lying  at  the  foundation  of  the  whole  course 
of  reasoning  that  gives  rise  to  the  idea  of  mathematical 
probability. 

31 


32  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

The  principle  is  this.  //  a  number  of  different  events 
are  equally  possible  as  regards  constant  conditions  (that 
is,  if  there  is  no  persistent  reason  why  one  should  occur 
rather  than  another),  and  all  are  repeatedly  given  oppor- 
tunity to  occur,  they  will  in  the  long  run  occur  with 
equal  average  frequcTicy.  The  same  principle  may  be  ex- 
pressed by  saying  that  if  we  observe  events  occurring  with 
equal  frequency,  we  conclude  that  the  constant  conditions 
under  which  they  occur  are  uniform. 

The  principle  is  well  illustrated  by  the  throwing  of  dice. 
If  a  die  is  exactly  cubical,  of  homogeneous  material 
(not  "  loaded  ")  and  the  spots  do  not  shift  the  center  of 
gravity  to  one  side,  and  if  it  be  cast  a  great  number  of 
times  absolutely  at  random,  each  face  will  come  up,  on 
the  average,  one  throw  out  of  six.  (Of  course  these  ideal 
conditions  are  not  realized  in  practice.) 

We  are  so  accustomed  to  the  operation  of  this  law  of 
probability  in  daily  experience  that  it  is  taken  as  a 
matter  of  course,  like  the  force  of  gravitation;  yet  its 
existence  is  really  a  mystery.  We  are  here  obliged  to 
admit  that  there  is  a  law  controlling  the  operations  of 
chance,  —  the  one  thing  that  would  seem  to  obey  no  law. 

16.  Definition  of  Mathematical  Probability.  —  Definite 
numerical  significance  may  now  be  given  to  the  probability 
of  occurrence  of  certain  classes  of  events. 

If  an  event  may  occur  in  a  equally  possible  ways, 
and  at  the  same  time  b  equally  possible  alternatives 
are  presented  in  all  (including  the  a  ways  in  which  the 


ON    PROBABILITIES  33 

event  may  happen),  then  the  probability  of  the  event  in 
question  is  defined  as  the  ratio 

V  =  \  (3) 

That  is,  there  are  a  chances  favoring  the  event  out  of  a 
total  of  b  possible  chances ;  and  according  to  the  principle 
set  forth  above,  if  a  great  number  of  trials  are  made,  the 
event  does  happen,  on  the  average,  a  times  out  of  b. 

As  an  example,  let  us  express  the  probability  of  draw- 
ing an  ace  from  a  deck  of  fifty-two  playing  cards,  the  draw- 
ing being  done  absolutely  at  random.  Any  one  of  the 
fifty-two  cards  may  be  drawn,  so  that  the  total  num- 
ber of  alternatives  is  fifty-two.  An  ace  may,  however, 
be  drawn  in  only  four  ways,  viz.,  by  drawing  the  ace 
of  spades,  the  ace  of  clubs,  the  ace  of  hearts  or  the 
ace  of  diamonds.  Here,  then,  &  =  52,  a  =  4,  and  the 
probability  of  drawing  an  ace  is  ^^,  or  -^-^. 

What  would  be  the  probability  of  drawing  a  red  ace? 
Of  drawing  the  axie  of  diaTnonds  ? 

All  problems  in  probability  may  be  solved  by  the  appli- 
cation of  the  definition  expressed  in  equation  (3).  But 
such  direct  application  would  be  very  diflOicult  in  the  more 
complicated  cases,  and  special  rules  and  formulas  are 
therefore  to  be  devised  wliich,  when  properly  classified 
and  applied,  greatly  simplify  such  problems. 

From  the  definition,  it  follows  that  probability  is  a 
purely  numerical  ratio,  and  depends  upon  no  unit  of 
measure.  Moreover,  this  ratio  cannot  exceed  unity. 
The  probability  unity  would  denote  certainty,  since  if  an 


34     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

event  may  happen  in  n  ways,  and  only  n  alternatives  are 
possible,  the  event  must  happen.  From  this  it  follows 
that  if  the  probability  of  an  event  is  p,  the  probability 
of  its  failure  to  happen  is 

p'  =  1  -  p.  (4) 

For,  if  the  event  can  happen  in  a  ways  out  of  b,  it  can  fail 

to  happen  inb  —  a  ways  out  of  b,  the  probability  of  failure 

therefore  being 

b—a     -,      a     -, 

—  =i--=i-p. 

More  generally,  the  sum  of  the  probabiHties  of  all  possible 
alternatives  is  unity. 

The  probability  zero,  on  the  other  hand,  implies  im- 
possibility. It  may  be  interpreted  as  meaning  that  there 
is  no  way  for  the  event  to  happen,  i.e.,  a  =  0;  or  in  cases 
where  the  total  number  of  alternatives  is  infinite,  or  at 
least  extremely  large,  while  the  event  in  question  may 
happen  in  only  a  very  few  ways,  the  zero  or  infinitely 
small  probability  denotes  impossibility  or  at  most  only 
extremely  remote  possibility.  But  the  distinction  be- 
tween absolute  impossibility  and  the  case  in  which  the 
possibility  is  only  remote  is  of  some  importance,  as  will 
be  seen,  in  the  theoretical  discussion  of  the  distribution  of 
errors. 

17.  Permutations.  —  The  solution  of  problems  in  prob- 
ability involves  the  determination  of  the  number  of  ways 
in  which  an  event  can  occur,  as  well  as  the  total  number  of 
possible  alternatives.     In  very  simple  cases  this  may  be 


ON    PROBABILITIES  35 

done  by  inspection.  For  example,  if  one  is  expecting  the 
arrival  of  three  different  persons,  A,  B,  C,  it  is  easy  to 
determine  the  probability  of  their  coming  in  the  order 
named.  There  are  obviously  six  different  orders  in 
which  they  may  come ;  namely,  ABC,  ACB,  BAC,  BCA, 
CAB,  CBA.  The  probability  of  their  coming  in  the  order 
ABC  is  therefore  J.  But  let  there  be  a  hundred  persons 
instead  of  three,  and  the  number  of  orders  becomes  so 
enormous  as  to  be  unmanageable  by  inspection.  We 
must  then  resort  to  the  use  of  general  formulas. 

The  linear  permutations  of  a  number  of  things  are  the  dif- 
ferent ways  in  which  the  things  may  be  arranged  in  a  row, 
or  in  which  they  may  occur  in  order  of  time.  There  are, 
for  example,  six  linear  permutations  of  the  letters  A,  B,  C. 

There  is  a  general  expression  for  the  number  of  permuta- 
tions of  Q  different  things,  derived  easily  by  the  following 
reasoning.  Of  one  thing,  there  is  evidently  but  one  per- 
mutation. Of  two  things,  since  either  may  come  first, 
there  are  two  permutations.  Of  three  things,  any  one 
may  come  first,  and  with  a  given  one  coming  first,  there 
are  two  arrangements  of  the  two  remaining;  therefore 
the  number  of  permutations  of  three  things  is  3  X  2  =  6. 
For  four  things,  by  the  same  reasoning,  the  number  is 
4  X  3  X  2  =  24.  And  in  general,  the  number  of  per- 
mutations of  Q  things  is 

Pq  =  Q(Q-1)(Q-2)-3.2.1=Q!  (5) 

We  have  here  assumed  that  none  of  the  Q  things  are 
duplicates.     Let  us  now  take  a  case  where   there   are 


36     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

duplicates,  as  in  the  group  of  letters  AABBBCCCC.  If 
we  distinguish  between  the  different  A's,  etc.,  the  case  is, 
of  course,  the  same  as  if  the  letters  were  all  different. 
But  if  we  consider  one  A,  for  example,  the  same  as  another, 
and  permute  without  regard  to  which  of  them  is  being 
used  in  a  particular  place,  the  number  of  permutations 
is  less.  It  will  be  easy  for  the  student  to  show  as  an 
exercise  that  if  there  are,  in  a  number  of  things,  m  of  one 
kind,  n  of  another,  r  of  another,  s  of  another,  etc.,  the  total 
number  being  Q  =  m  -]-  n  +  r  -\-  s  -\-  •••,  the  number  of 
distinguishable  permutations  of  the  Q  things  is  -, 

ml  nl  rl  Si  ••• 

Thus  for  the  above  set  of  nine  letters,  of  which  two  are 
A's,  three  B's  and  four  C's,  the  number  is 

p  (2.3.4)^  — ^J —  =  1260. 
2!3!4! 

If  there  is  only  one  thing  of  a  kind  in  the  group,  so  that 
m,  n,  .' .  .  are  each  unity,  (6)  becomes  equivalent  to  (5). 

18.  Combinations.  —  The  different  groups  which  can 
be  formed  from  a  number  of  things,  taken  so  many  at  a 
time,  are  called  combinations.  The  different  combinations 
of  the  three  letters  A,  B,  C  taken  two  at  a  time  are  AB, 
AC,  BC. 

If  we  further  take  into  account  the  possible  permuta- 
tions of  each  combination,  we  have  what  may  be  called 
the  permuted  combinations  of  the  series  of  things  considered. 


ON   PROBABILITIES  37 

Thus  the  permuted  combinations  of  A,  B,  C  are  AB,  BA, 
AC,  CA,  BC,  CB.  It  is  easier  to  derive  first  the  general 
formula  for  the  number  of  permuted  combinations. 

Let  the  number  of  permuted  combinations  of  Q  things 
taken  n  at  a  time  be  designated  by  the  symbol  PCq^^\  If 
they  are  taken  two  at  a  time,  any  one  of  the  Q  things  may 
be  taken  as  the  first,  and  any  one  of  the  Q  —  I  remaining 
things  may  be  taken  as  the  second,  so  that 

If  taken  by  threes,  any  one  of  the  Q  (Q  —  1)  permuted 
combinations  of  two  each  may  constitute  the  first  two, 
followed  by  any  one  of  the  Q  —  2  remaining  things  as 
the  third.     Then 

By  continuing  the  same  reasoning  until  there  are  n  things 
taken  at  a  time,  we  readily  deduce 

PCq^-^=Q{Q  -DiQ-  2)  ...  to  n  factors.         (7) 

If  n  =  Q,  this  becomes  identical  with  (5),  since  all  the  things 
are  permuted  at  once. 

To  express  now  the  number  of  combinations  of  Q  things 
taken  n  at  a  time,  without  regard  to  their  arrangement,  it 
is  necessary  only  to  note  that  the  PCq^""^  permuted  combina- 
tions include  not  merely  those  made  up  of  different  things, 
but  all  the  permutations  of  each  of  the  groups  of  n  different 
things.     Since  n  things  are  permuted  in  n !  different  ways 

(5),  there  are  only  — -  as  many  combinations  as  permuted 
nl 


38     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

combinations.     That  is, 

^  M^QjQ  -  1)  •••  to  n  factors  .^. 

n  ! 

As  an  illustration  of  this  problem,  let  us  find  how  many 
different  hands  at  whist,  each  made  up  of  thirteen  cards, 
could  be  drawn  from  a  pack  of  fifty-two  cards.  Here 
Q  =  52,  n  =  13,  and  the  solution  is 

^^^(13)  ^  52 -5^1 -50  ♦♦^■40  ^  635^013,559,600. 

Then  the  probability  of  drawing  any  one  specified  hand  is, 
by  definition,  the  exceedingly  small  reciprocal  of  this 
number. 

As  a  final  problem  in  combinations,  let  there  be  s  series 
of  things,  the  number  of  things  in  the  respective  series 
being  Qi,  Q2,  •••,  §«;  to  determine  how  many  different 
combinations  can  be  formed  by  taking  one  thing  from 
each  series. 

.  The  number  of  combinations  of  two  each  which  can  thus 
be  formed  from  the  first  two  series  is  Q1Q2,  since  each  of 
the  Qi  things  in  the  first  series  can  be  successively  com- 
bined with  each  of  the  Q2  things  in  the  second.  Bringing 
in  now  the  third  series,  each  of  the  Q1Q2  combinations  just 
considered  may  be  combined  with  each  of  the  Q3  members 
of  the  third  series,  making  Q1Q2Q3  combinations ;  and  so 
on.  Clearly,  then,  the  number  of  combinations  that  can 
be  so  formed  from  the  s  series  is  the  product 

.CQ,..Q,  =  (iiQ2Q3-Q.^  (9) 


ON   PROBABILITIES  39 

For  example,  let  there  be  three  series  of  letters : 
A,    B,    C, 
A2    B2    C2 
A,    B, 
A, 

The  number  of  combinations  of  the  form  ABC  that  can 
be  selected  from  them  is  4  X  3  X  2  =  24.  Let  the  stu- 
dent write  these  combinations. 

19.  Probability  of  Either  of  Two  or  More  Events.  —  If 
the  probability  of  an  event  A  is  pa,  that  of  an  event  B  is 
Pby  that  of  an  event  C  is  pc,  etc.,  then  it  is  easy  to  show- 
that  the  probability  that  one  or  another  of  these  events  will 
happen  is  pa  -\-  Pb  +  Pc  -\-  -",  it  being  understood  that 
only  one  of  these  events  can  happen.  For,  suppose  the 
event  A  may  happen  in  a  ways,  the  event  B  in  6  ways,  etc., 
and  that  the  total  number  of  alternatives  is  T.  (In 
general,  T  will  be  greater  than  the  sum  of  a,  b,  etc. ;  that  is, 
it  is  not  necessary  that  any  one  of  the  events  A,  B,  etc. 
shall  happen.)  Then  by  definition,  the  probabilities  of  the 
respective  events  are 

a  b      . 

Va^-^,     P6  =  ^.  etc. 

If  we  designate  by  X  the  event  of  some  one  of  the  events 
A,  B,  etc.  happening,  without  specifying  which,  then, 
since  the  number  of  ways  in  which  X  can  occur  is  a  +  6 
+  ••*,  the  probability  of  X  is 

Px=  rr =Va-\-Vh-\-   ••••  (10) 


40  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

As  an  example,  let  there  be  in  a  bag  three  balls  of  iron,  two 
of  glass,  five  of  wood,  seven  of  lead,  six  of  rubber,  one  of 
ivory  and  four  of  copper,  and  let  one  be  drawn  at  random. 

The  probability  that  a  metal  ball  will  be  drawn  is  then 
A  H"  A  +  2*?  =  i*  since  a  metal  ball  is  drawn  if  the  re- 
sult be  an  iron  ball,  a  lead  ball  or  a  copper  ball. 

This  principle  of  additive  probabilities  for  alternative 
events  is  made  use  of  in  estimating  premiums  on  so-called 
"  joint  "  life  insurance  policies. 

20.  Probability  of  the  Concurrence  of  Independent 
Events.  —  Quite  a  different  problem  is  that  of  finding  the 
probability  that  all  of  a  specified  set  of  independent  events 
shall  occur.  As  before,  designate  the  respective  events  by 
A,  B,  C,  etc.,  their  respective  separate  probabilities  by 
Pa,  Pb,  etc. ;  and  designate  the  event  of  their  all  occurring 
by  Z.  Suppose  the  event  A  may  occur  independently  in 
a  ways  out  of  a  alternatives,  B  in  6  ways  out  of  /3  alter- 
natives, etc.,  so  that 

a  h      . 

Pa  =  -,     pb  =  -,  etc. 
a  p 

It  is  of  course  understood  that  when  all  the  events  A,  B,  C, 
etc.,  are  given  opportunity  to  happen,  some  one  of  the  a 
alternatives  connected  with  A  will  happen,  some  one  of  the 
j8  alternatives  connected  with  B  ivill  happen,  etc.,  but 
that  only  one  of  each  can  happen.  The  total  number  of 
possible  outcomes  is  therefore  the  number  of  combinations 
that  can  be  formed  by  selecting  one  from  each  group  of 
alternatives,  namely,  the  product  ajSy  •••   (9).     Likewise, 


ON   PROBABILITIES  41 

the  number  of  different  ways  in  which  the  events  A,  B, 
etc.,  can  all  occur  is  the  product  abc  •••.  It  follows  that 
the  probability  of  all  occurring,  that  is,  the  probability 
of  the  event  Z,  is 

abc  '"abc  .^  ^. 

Vz  =  — =  -   '-'-•"  =PaPbPc  ••••  (11) 

apy  •••       a      p      y 

That  is  to  say,  the  probability  of  the  concurrence  of  two  or 
more  independent  events  is  the  product  of  the  probabilities 
of  the  respective  events  considered  separately.  This  product 
is  of  course  less  than  any  one  of  its  factors. 

To  make  the  meaning  of  this  clear,  suppose  that  it  is 
known  that  a  person  A  will  spend  five  hours  in  a  certain 
place  between  6  a.m.  and  6  p.m.,  and  that  another  person 
B  will  spend  three  hours  there  during  the  same  interval, 
but  nothing  is  known  as  to  when  these  hours  will  be.  If 
we  visit  the  place  at  any  random  moment,  the  probability 
of  finding  A  there  at  that  moment  is  i\ ;  the  probability  of 
finding  B  there  at  that  moment  is  y2  .  Then  the  probability 
of  finding  them  both  there  at  that  moment  is  i\  X  i%  =  ^\. 
But  the  probability  of  finding  either  A  or  B  there  is 
■^2  +  -12  =  I-  Let  the  student  analyze  this  problem  more 
closely,  showing  how  the  values  stated  for  the  probabilities 
can  be  deduced  from  the  definition  of  probability. 

21.  The  Coin  Problem.  —  Suppose  that  the  result  of 
an  experiment  may  be  either  one  of  two  things,  A  and  B^ 
which  are  equally  likely  to  occur,  and  that  the  result  must 
be  one  or  the  other,  but  cannot  be  both.  The  probability 
of  either  result  is  then  J.     Let  us  determine  what  is  the 


42  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

probability,  if  the  experiment  be  performed  Q  times,  that 
it  will  result  n  times  one  way  and  Q  —  n  times  the  other. 
A  coin  tossed  at  random  illustrates  the  problem ;  for 
example,  if  it  be  tossed  a  hundred  times,  what  is  the  prob- 
ability that  it  will  turn  up  heads  thirtj^-eight  times  and 
tails  sixty-two  times?  Here  Q  =  100,  n  =  38  (or  62). 
The  required  probability  is  a  function  of  n ;  and,  further- 
more, it  is  evidently  the  same  function  oi  Q  —  n  that  it  is 
of  n. 

The  first  thing  to  determine  is,  in  how  many  ways  the 
result  A  may  happen  n  times  out  of  Q.  In  100  throws  of 
the  coin,  heads  may  come  up  38  times  and  tails  62  times  in 
a  large  variety  of  ways :  for  example,  1  H.,  2  T.,  37  H. 
and  60  T.,  in  order,  would  fulfill  the  condition ;  or,  equally 
well,  8  H.,  5  T.,  30  H.,  and  57  T.  The  number  of  ways 
in  which  A  may  happen  n  times  and  B,  Q  —  n  times  is 
readily  seen  to  be  equal  to  the  number  of  distinguishable 
permutations  of  Q  things,  n  being  of  one  kind  and  Q  —  n 
of  the  other  (Art.  17),  which  is 

Pq^^^^-^^=     .    ,^'     .,■  (12) 

71 !   {Q-n)l 

Or,  it  is  equal  to  the  number  of  combinations  of  Q  things 
taken  n  at  a  time,  since  out  of  the  totality  of  Q  events, 
the  n  events  A  may  be  selected  wherever  desired.  Hence 
another  expression  for  the  required  number  is  equation 
(8),  which  the  student  may  readily  show  to  be  equivalent 
to  (12).     We  shall  use  equation  (12). 

Next  we  must  determine  the  total  number  of  possible 


ON   PROBABILITIES  43 

alternatives.  This  may  be  done  by  adding  together  the 
values  of  the  expression  (12)  obtained  by  giving  n  all 
integral  values  from  0  to  Q.    These  are  tabulated  below. 

n 
0 
1 


p^(«,  Q-n) 
1 

Q 

QXQ-i) 

Q(Q 

2! 
-iXQ-2) 

3! 
QiQ-i) 

2! 

Q 
1 

Q-2 

Q-i 
Q 

The  expressions  obtained  for  Pq(^'<?-")  as  n  varies  from 
0  to  §  are  at  once  seen  to  be  the  successive  coefficients  of 
the  expansion  of  a  binomial  with  exponent  Q,  and  their 
sum  is  therefore  equal  to  2^.     That  is. 

We  now  have  the  two  elements  of  the  solution  of  the 
coin  problem,  namely,  the  number  of  ways  in  which  event 
A  can  happen  n  times  and  event  B,  Q  —  n  times,  given 
by  (12),  and  the  total  number  of  alternatives,  given  by 
(13).  The  required  probability  of  the  specified  outcome 
is  therefore 

^"•«-"=n!(Q-l)!2«-  ('*) 


44  THEORY   OF   ERRORS   AND   LEAST   SQUAR^ES 

Thus  the  probabiHty  of  the  result  38  heads  and  62  tails  is 

100! 
^'''''     38!   62!  2100- 

The  reason  for  introducing  the  coin  problem  will  appear 
later. 

22.  Important  Exercise.  —  It  is  now  very  desirable, 
for  the  purpose  in  hand,  that  the  student  faithfully  per- 
form the  following  exercise.  Suppose  a  coin  tossed  ten 
times.  Find  the  probability  of  each  of  the  following 
possible  results : 

n     '  10  —  n  n  10— n 

10  heads  and  0  tails  4  heads  and    6  tails 

9  heads  and  1  tails  3  heads  and    7  tails 

8  heads  and  2  tails  2  heads  and    8  tails 

7  heads  and  3  tails  1  heads  and    9  tails 

6  heads  and  4  tails  0  heads  and  10  tails 
5  heads  and  5  tails 

Considerations  of  symmetry  will  shorten  the  work.  Now 
plot  a  series  of  points,  of  which  the  abscissas  shall  represent 
the  quantity  n  —  5  (n  being  the  assumed  number  of  heads) 
and  the  ordjnates  the  computed  probabilities  of  the  re- 
spective results,  using  a  convenient  scale  for  each.  Does 
the  resulting  curve  resemble  any  other  curve  that  has 
hitherto  come  to  your  notice  ?  Test  the  theory  by  actual 
experiment  with  a  coin,  or  better,  a  flat  bone  disk,  record- 
ing the  outcome  of  every  ten  throws.  This  exercise,  if 
carefully  performed  and  studied,  will  assist  the  student 


ON   PROBABILITIES  45 

to  a  much  better  understanding  of  the  behavior  of  error 
distribution  than  he  could  attain  without  it. 

23.  Empirical  or  Statistical  Probability.  —  As  has  been 
before  noted,  in  the  majority  of  the  events  of  Kfe,  the  con- 
ditions are  far  too  compHcated  to  admit  of  any  such  analy- 
sis as  has  been  applied  to  the  problems  concerning  cards, 
balls,  coins,  etc.  But  it  may  happen  that,  when  the  con- 
ditions are  sufficiently  constant  throughout  a  long  series 
of  observations,  the  probability  of  such  a  complex  event 
may  be  deduced  from  the  observed  results.  It  is  upon 
this  principle  that  reliance  is  placed  upon  statistics.  As 
a  very  important  example,  we  cannot  compute,  by  any 
theoretical  formula,  the  probability  that  a  person  ten  years 
old  will  live  to  be  sixty.  But  if  the  statistics  show  that 
out  of  every  100,000  persons  ten  years  of  age,  58,000  do 
live  to  be  sixty,  we  may  conclude  that  the  required  prob- 
ability is  0.58.  In  a  similar  manner  it  has  been  deter- 
mined that  the  probability  that  a  person  sixty  years  old 
will  live  one  more  year  is  0.97,  since  97  per  cent,  of  those 
attaining  the  age  of  sixty  do  live  another  year.  The  im- 
portance of  such  knowledge,  and  its  bearing  on  the  practi- 
cal problems  of  the  world,  such  as  life  insurance,  are 
self-evident. 

EXERCISES 

24.  1.  What  is  the  probability  of  throwing  a  six  in 
two  throws  of  a  single  die?  In  a  single  cast  of  two 
dice? 


46     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

2.  How  many  possible  arrangements  are  here  of  the 
letters  in  the  word  travel?  How  many  distinguishable 
arrangements  of  the  letters  in  the  word  minimum  f 

3.  A  hostess  wishes  to  have  as  guests  the  same  number 
of  ladies  as  gentlemen.  She  has  planned  to  have  the 
guests  find  their  partners  by  the  matching  of  colored 
ribbons,  each  guest  wearing  two  colors,  there  being  but  five 
different  colors  in  all.  How  many  guests  may  she  invite  ? 
Would  she  be  able  to  distinguish  more  couples  by  giving 
each  guest  three  colors?     Four  colors? 

4.  How  many  different  football  elevens  could  be  formed 
from  a  squad  of  fifteen  players?  What  chance  would 
any  one  player  have  of  getting  on  a  picked  eleven  if  it 
were  chosen  by  lot? 

^  5.  In  a  certain  organization  there  are  two  candidates 
for  an  office  and  thirty  voters.  What  is  the  probability 
that  there  will  be  a  tie?  That  either  candidate  will  re- 
ceive a  majority  of  exactly  f  ? 

6.  By  measuring  a  number  of  ordinates  of  the  curve 
obtained  from  the  target  experiment  (Art.  11),  determine 
empirically  the  relative  probabilities  of  the  respective 
errors  in  aim. 

7.  Find  the  number  of  combinations  of  three  things  in 
eight;   the  number  of  permuted  combinations. 

8.  A  student  council  is  to  be  made  up  of  five  members 
from  each  of  four  college  classes,  whose  respective  member- 
ships are  150,  105,  75  and  56.  In  how  many  different 
ways  may  the  council  be  made  up  ? 


ON   PROBABILITIES 


47 


9.  Three  things  are  selected  at  random  from  eight, 
then  returned ;  and  then  another  random  selection  of 
three  is  similarly  made.  What  is  the  probability  that 
the  two  selections  will  be  exactly  reverse  permutations 
of  the  same  three  things  ? 

10.  A  new  janitor  has  a  bunch  of  twenty-eight  nearly 
similar  keys,  one  for  each  door  of  the  building.  What 
is  the  probability  of  his  being  able  to  unlock  the  first 
three  doors  with  only  one  trial  each?  Solve  also  on 
the  supposition  that  he  marks  the  keys  as  he  discovers 
them. 

11.  What  is  the  probability  that  a  whole  number  of 
four  figures,  selected  at  random,  will  have  two  figures 
alike  and  the  other  two  figures  alike  ? 

12.  The  following  data  are  taken  from  the  American 
Experience  Mortality  Tables  used  by  life  insurance  com- 
panies in  computing  risks. 

Out  of  100,000  persons  ten  years  of  age, 


100,000  1 
92,637  I 
89,032  1 
85,441  1 
81,822  1 
78,106  1 
74,173  1 
69,804  1 
64,563  1 
57,917  1 


ve  to  be  at  least  10 
ve  to  be  at  least  20 
ve  to  be  at  least  25 
ve  to  be  at  least  30 
ve  to  be  at  least  35 
ve  to  be  at  least  40 
ve  to  be  at  least  45 
ve  to  be  at  least  50 
ve  to  be  at  least  55 
ve  to  be  at  least  60 


48     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

49,341  live  to  be  at  least  65 
38,569  live  to  be  at  least  70 
Plot  a  curve  representing  these  data. 

13.  Find  your  own  chance  of  living  to  be  at  least  seventy 
years  old,  using  the  curve  in  Ex.  12. 

14.  Three  men  are  respectively  30,  27  and  22  years  old. 
Find  the  probability  that  they  will  all  live  to  be  60  or  over. 

15.  Two  brothers  are  respectively  25  and  35  years  old, 
and  their  father  is  60.  The  elder  is  to  inherit  the  estate 
if  living  at  the  father's  death,  otherwise  the  younger  will 
inherit  it ;  and  at  the  death  of  the  elder  son,  the  younger 
will,  if  living,  inherit  the  estate  from  him.  Find  the  prob- 
ability that  the  elder  son  will  own  the  estate  five  years 
hence ;  that  the  younger  will  own  it  ten  years  hence. 


CHAPTER  IV 

THE   ERROR   EQUATION   AND   THE   PRINCn>LE   OF 
LEAST   SQUARES 

25.  Analogy  of  Error  Distribution  to  Coin  Problem.  — 
It  was  pointed  out  in  Art.  5  that  an  error  in  measurement 
is  the  resultant  of  innumerable  small  disturbances  of 
different  kinds,  the  presence  of  many  of  which  may  not 
be  even  suspected.  These  disturbances  operate,  some  in 
one  way,  some  in  the  other ;  that  is,  some  tend  to  produce 
positive  error  and  some  negative.  The  resultant  error 
depends  on  the  1-elation  of  the  number  of  positive  disturb- 
ances to  the  number  of  negative  disturbances.  If  nearly 
all  are  positive,  the  error  will  be  positive  and  large;  if 
nearly  all  are  negative,  a  large  negative  error  will  result ; 
while  if  about  the  same  number  are  positive  as  negative, 
the  error  will  be  small.  This  does  not  imply  that  the  dis- 
turbances are  all  of  the  same  magnitude.  By  way  of  illus- 
tration, suppose  we  select  from  a  sand-heap,  at  random, 
a  thousand  grains  of  sand,  and  put  eight  hundred  of  them 
on  the  left  pan  of  a  balance  and  two  hundred  on  the  other. 
There  is  hardly  a  remote  possibility  that  the  former  will 
not  very  largely  overbalance  the  latter.  But  if  we  put 
five  hundred  on  each  pan,  there  will  be  little  preponder- 
ance one  way  or  the  other.  And  this  does  not  imply,  by 
E  49 


50     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

any  means,  that  the  grains  are  all  equal  in  weight ;  some 
individual  particles  may  be  ten  times  heavier  than  others. 

Now  there  is  a  remarkable  and  useful  analogy  between 
the  theory  of  error  distribution  and  the  so-called  coin 
problem  (Art.  21),  an  analogy  that  the  student  has  no 
doubt  already  observed.  It  is  easily  deduced  that  the 
most  probable  result  of  a  number  of  throws  of  a  coin  is 
that  they  will  be  half  heads  and  half  tails.  In  general, 
this  normal  result  will  be  departed  from  in  greater  or  less 
degree,  so  that  in  one  hundred  throws  we  frequently  ob- 
tain fifty-five  heads  and  forty-five  tails,  or  less  frequently, 
sixty  heads  and  forty  tails,  etc.  This  departure  from  the 
normal  or  most  probable  result  may  be  looked  upon  as 
a  sort  of  error.  Like  an  error  in  measurement,  it  is  com- 
plex in  character,  depending  upon  the  result  of  each  in- 
dividual throw.  Each  throw,  head  or  tail,  affects  the 
final  outcome  one  way  or  the  other,  just  as  each  small 
disturbance,  positive  or  negative,  affects  the  result  of  an 
observation  in  measurement.  A  little  consideration  of 
the  two  cases  will  bring  out  their  analogy  quite  clearly. 

We  are  therefore  justified  in  assuming  that  the  proba- 
bility of  the  occurrence  of  an  error  is  a  function  of  the 
magnitude  of  the  error  in  much  the  same  manner  as  the 
probability  of  a  departure  from  the  half-and-half  result 
in  tossing  the  coin  is  a  function  of  the  extent  of  the  de- 
parture. It  is,  in  fact,  upon  this  line  of  reasoning  that 
Hagen's  deduction  of  the  error  equation  is  based.  The 
deduction  is,  however,  rather  cumbersome,  and  we  shall 
follow  instead  the   more  elegant  method  due  to  Gauss. 


THE  ERROR  EQUATION  51 

It  may  be  remarked  here  that  the  theory  of  departures 
is  a  very  general  one  and  finds  apphcation  in  a  large  variety 
of  problems  of  common  experience,  such  as  the  distribution 
of  shots  on  a  target  and  the  distribution  of  given  charac- 
teristics among  the  members  of  a  biological  group. 

26.  The  Most  Probable  Value  from  a  Series  of  Direct 
Measurements.  The  Arithmetical  Mean.  —  If  a  series 
of  measurements  be  made  upon  a  single  quantity  under 
as  nearly  constant  conditions  as  possible,  the  result  is, 
in  general,  a  series  of  different  values,  each  approximating 
the  true  value  of  the  measured  quantity.  No  one  of  them 
is  the  true  value,  however,  and  it  now  becomes  a  matter  of 
judgment  to  select,  from  all  possible  values,  such  a  one 
as  will  make  the  actual  distribution  of  the  results  appear 
most  natural.  An  analogous  case  would  be  this  :  Suppose 
that  after  all  the  shots  had  been  fired  at  the  target  in  the 
first  exercise  of  Art.  11,  the  central  line  aimed  at  were 
erased,  and  we  were  required,  from  the  given  distribution 
of  the  shots,  to  judge  as  to  where  the  line  had  been ;  we 
could  do  no  better  than  to  select  a  position  that,  from  the 
concentration  of  shots  about  it  and  their  symmetry  with 
respect  to  it,  seems  to  be  the  most  probable  one.  Likewise, 
in  a  series  of  measurements,  we  are  aiming  at  a  true 
value,  the  most  probable  location  of  which  can  only  be 
estimated  by  an  examination  of  the  distributed  results. 

The  symmetry  of  the  distribution  of  errors  in  cases  where 
the  true  value  is  known,  as  also  in  the  analogous  coin  and 
target  problems,  leads  at  once  to  the  common  axiom  of 


52     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

experience,  that  the  best  value  to  adopt  in  the  ease  of  a 
series  of  direct  observations  on  a  single  quantity  is  the 
arithmetical  mean  or  average  of  the  observations.  If  the 
several  measured  results  be  designated  by  Si,  S2,  "',Sn, 
and  their  mean  be  m,  then  the  residuals  (Art.  7)  are  re- 
spectively 

pi  =  si  —  m, 

P2  =  S2  —  m, 


Pn  =  Sn  -  m. 

Adding  these  we  obtain 

2/3  =  25  -  nm  =  0,       .  (15) 

which  expresses  the  fact  that  the  arithmetical  mean  of 
the  results  is  the  value  with  respect  to  which  they  are  sym- 
metrically placed,  the  algebraic  sum  of  the  differences 
being  then  equal  to  zero ;  and  that  therefore  this  mean  is 
the  most  probable  value  that  can  be  assumed. 

27.  Gauss's  Deduction  of  the  Error  Equation.  —  Let  q 
represent  the  unknown  true  value  of  a  quantity  and  let  a 
series  of  n  measurements  be  made  upon  it,  the  number  n 
being  supposed  very  large.  Let  the  errors  arising  from 
the  respective  measurements  be  Xi,  X2,  •••,  Xn.  It  has 
been  seen  that  the  probability  of  the  occurrence  of  an 
error  is  some  sort  of  inverse  function  of  its  magnitude. 
Designating  the  probabilities  of  these  respective  errors 
Kv  2/ij  2/2>  •••>  Vnt  this  fact  may  be  expressed  by  the 
equations 


THE  ERROR   EQUATION  53 

It  is  the  form  of  this  function  f(x)  that  we  are  seeking  to 
determine. 

Now,  as  above  noted,  we  do  not  know  the  true  value  q 
of  the  observed  quantity,  and  therefore  we  do  not  know 
the  true  errors  x.  We  may  however  assume  various  tenta- 
tive values  for  q  and  study  the  resulting  tentative  systems 
of  errors,  particularly  with  a  view  to  selecting  that  one 
which  seems  most  naturally  distributed,  in  accordance 
with  the  notions  of  error  distribution  that  experience  has 
taught  us.  In  this  sense,  therefore,  we  may  think  of  q 
and  the  errors  x  as  variables  subject  to  our  control,  and  the 
probabilities  y  will  then  vary  accordingly.  With  this 
understanding,  then,  we  are  seeking  to  find  that  system 
of  values  for  the  a:'s  which,  as  a  whole,  has  the  greatest 
probability. 

If  the  outcome  of  a  series  of  measurements  be  the  sys- 
tem of  errors  Xi,  X2,  •••,  Xn,  this  result  may  be  looked  upon 
as  the  concurrence  of  n  independent  events,  each  of  which 
is  the  obtaining  of  one  of  the  errors  x.  Then  according 
to  Art.  20,  the  probability  of  this  outcome,  designated  by 
y,  is  the  product  of  the  probabilities  of  the  separate  errors, 
namely 

Y  =  yiy2  -  2/n  =  /fe)  -/fe)  -fixn).  (16) 

In  order,  therefore,  that  the  system  of  a:'s  shall  have  the 
greatest  probability,  as  required,  the  value  assumed  for  q 


54     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

should  be  such  that  the  expression   F  is  a  maximum; 
which  condition  will  be  attained  when 

dY 

^  =  0.  (17) 

dq 

Differentiating  (16) 

dY^^_  ^  dJM_^_y_  .  dfjx,)  ^    ^^ 
dq     f{xi)         dq        fix^)         dq 


Y      .  df{Xn)  _  Q 


f{xn)         dq 
or  y\df{x,)  ^df{x2)  I   ,,     J  dfjxjl 

dqLfiXi)  f(X2)  f{Xn)J 

=  T\.d  log  Rxi)  +d  log  f{x2)  +  ...  +  ^  log  f(Xn)]  =  0. 
dq 

Now  let  d  \ogf{x)  =  (f)(x)dx,  where  <j)  is  another  unknown 
function  of  x,  thus  simply  related  to  /.  Then  canceling 
out  the  Y, 

<t>{xi)^  +  <i>(x2)^+^-  +  cf>M^=0.         (18) 
dq  dq  dq 

If  the  results  of  the  respective  n  measurements  on  q  be 
designated  by  ^i,  ^2,  •••,  ^n,  having  definite,  fixed  values, 
then  the  errors  x  are  (Art.  7) 

Xi  =  si-  g, 

X2=S2-  q, 


THE  ERROR   EQUATION  55 

from  which,  at  once, 

dq       dq  dq 

(18)  then  reduces  to 

0(^1)  +  0(^2)  4-  •  •  •  +  <t>{xj  =  0.  (20) 

We  already  understand  enough  of  the  law  of  error  dis- 
tribution to  know  that  when  the  number  of  observations  is 
very  large,  the  number  of  positive  errors  of  given  magni- 
tude about  equals  the  number  of  negative  errors  of  the 
same  magnitude,  and  that  therefore  the  algebraic  sum  of 
the  errors  is  approximately  zero.  Since  in  our  theoretical 
discussion  the  number  of  observations  is  indefinitely 
large,  we  may  write,  therefore,  as  another  condition 
fulfilled  by  the  errors, 

Xi-\-X2-{-  •••  -\-Xn  =  0.  (21) 

It  now  remains  to  deduce  from  the  two  equations  (20)  and 
(21)  the  form  of  the  function  <f>,  from  which  the  original 
function  /  may  then  be  obtained.  It  is  not  difficult  to 
see  that  the  equations  are  satisfied  if 

</)(xi)  =  Kxi, 

<t>{X2)    =   KX2,         ' 


0(a:„)  =  Kxr., 

where  X  is  a  constant.  A  mathematical  proof  that  this  is 
the  necessary  relation  is  given  in  Note  A,  Appendix,  being 
omitted   here   to   avoid   distracting   attention   from  the 


56     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

main  problem.     We  may  write,  then, 

<t>{x)  =  Kx;  (22) 

or  since  <f>{x)dx  =  d  log  f{x)  =  d  log  y, 

d\og  y  =  Kxdx. 

Integrating,  log  1/  =  i  Kx^  +  c', 

or  y  =  e^^^^+<^.  (23) 

This  is  one  form  of  the  error  equation. 

The  expression  may,  however,  be  so  modified  as  to  ex- 
hibit the  relation  to  better  advantage.  We  have  seen 
that  the  larger  the  error,  the  less  likely  it  is  to  occur : 
the  larger  x  is,  the  smaller  is  y.  Clearly,  then,  K  must  be 
a  negative  quantity.  Replacing  ^  i^  by  —  h^,  and  e*^  by 
the  constant  c,  the  equation  assumes  the  more  usual  and 
more  useful  form 

y  =  ce-^''^\  (24) 

This  is  "the  most  important  equation  in  the  theory  of  errors, 
and  should  be  committed  to  memory. 

28.  Discussion  of  the  Error  Equation.  —  It  will  be 
interesting  to  examine  equation  (24)  to  see  how  closely 
the  law  of  error  thereby  expressed  agrees  with  the  conclu- 
sions already  reached. 

The  bilateral  symmetry  of  the  function  y  is  evident 
from  the  occurrence  of  x  in  the  second  degree  only.  This 
indicates  the  equal  probability  of  positive  and  negative 
errors  of  the  same  magnitude.     The  function  approaches 


THE  ERROR  EQUATION 


57 


zero  as  x  increases  in  magnitude ;  which  means  that  very 
great  errors  are  extremely  improbable.  The  derivatives 
of  the  function  are 

dy 


dx 


=  -2ch^xe-^'-\ 


dx^ 


(25) 
(26) 


From  these,  since  -^  =  0,  — ^  <  0  when  a:  =  0,  there  is  a 
dx  dx^ 

maximum  value  of  y  when  a:  =  0 ;  that  is,  the  error  zero 
has  the  greatest  probability. 

The  curve  shown  in  Fig.  5  represents  the  function, 
and  has  some  interesting  properties.  Its  symmetry, 
asymptotic  character 
and  central  maxi- 
mum merely  illus- 
trate what  has  just 
been  deduced  from 
the  equation.  The 
Y  intercept,  or  maxi- 
mum ordinate,  is  the 
quantity  c,  since  y  =  c 


d'y 


Fig.  5 


when  X  =  0.     If  we  put  ^  equal  to  zero,  which  is  the 

dx^ 

condition  for  points  of  inflection,  (26)  gives 
1  -  2  hV=  0, 
1 


Xi  =  ± 


h^J2 


(27) 


58     THEORY   OF   ERRORS   AND    LEAST   SQUARES 

This  is  the  distance  OD  or  OD',  corresponding  to  the  points 
of  inflection  P  and  P',  The  ordinate  of  these  points  is, 
by  substitution, 

2/i  =  +  -^,  (28) 

and  is  therefore  proportional  to  c. 

The  quantity  c  represents  the  probabiUty  of  the  error 
zero.  Now  the  probabihty  of  any  given  error  a:  is  a  function 
of  both  c  and  h,  since  it  changes  if  we  change  either  c 
or  h.  It  would  thus  appear  that  c  and  h  have  something 
to  do  with  the  precision  of  the  measurements,  and  that 
they  are  therefore  connected  with  each  other.  We  shall 
see  later  (Art.  54)  that  this  is  the  case,  and  also  that 
there  is  still  another  factor  in  the  probability  of  a  given 
error,  depending  upon  the  value  of  the  smallest  scale  in- 
terval in  terms  of  which  the  measurements  are  expressed. 

29.  The  Principle  of  Least  Squares  in  its  Simplest  Form. 
—  We  are  now  in  position  to  make  an  introductory  state- 
ment of  the  important  principle  which  gives  this  branch 
of  science  its  name,  —  the  principle  of  least  squares.  Be- 
fore we  are  through  with  the  theory  of  errors,  the  principle 
will  have  been  stated  several  times  in  successively  more 
complicated  forms,  as  the  problems  to  which  it  is  applied 
become  more  and  more  general.  So  far  we  have  been 
considering  only  the  simplest  case,  namely,  that  of  ob- 
servations of  equal  precision  upon  a  single  quantity; 
and  while  for  this  case  the  method  of  deducing  the  most 
probable  value  is  clear  without  reference  to  the  principle 


THE  ERROR  EQUATION  59 

of  least  squares,  still  it  will  be  interesting  and  instructive 
to  observe  how  the  assumption  of  the  arithmetical  mean 
as  the  most  probable  value  may  be  shown  to  be  in  accord- 
ance with  that  principle  in  the  simple  form  here  stated. 

The  simple  form  of  the  principle  referred  to  is  as  follows  : 

The  most  probable  value  of  a  measured  quantity  that  can 
be  deduced  from  a  series  of  direct  observations,  made  with 
equal  care  and  skill,  is  that  for  which  the  sum  of  the  squares 
of  the  residuals  is  a  minimum. 

The  law  governing  the  distribution  of  errors  has  already 
been  deduced  theoretically,  and  the  experience  of  number- 
less experimenters  testifies  to  its  truth.  We  have  there- 
fore a  right  to  expect  that,  when  we  have  made  a  long  series 
of  measurements  upon  a  single  quantity,  our  observations 
will  have  grouped  themselves  around  the  true  value  in 
a  manner  approximately  consistent  with  the  error  equation 
(24).  Then  it  is  logical  for  us  to  assume  a  value  for  the 
measured  quantity,  such  that  the  results  of  the  measure- 
ments will  be  so  grouped  with  respect  to  it.  This  is  the 
so-called  most  probable  value,  and  it  is  the  office  of  the  prin- 
ciple of  least  squares,  in  any  case,  to  point  out  the  way  of 
arriving  at  it. 

Let  the  results  of  the  n  observations  be  ^i,  ^2,  •••,  Sn- 
Then  if  we  designate  the  most  probable  value  sought  by 
m,  there  will  arise  a  corresponding  series  of  residuals 
Ph  P2,  •'•,  Pn,  each  of  which  is  found  by  subtracting  m 
from  the  corresponding  observation  s  (Art.  7).  If  m 
be  properly  chosen,  the  residuals  derived  from  it  will, 
like  true  errors,  be  found  to  be  distributed  in  accordance 


60     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

with  the  exponential  law  of  error  probability  (24),  so  that 
the  probabilities  of  the  respective  residuals  are 

2/2  =  ce-^'^, 
2/n  =  ce-^V. 

The  probability  of  the  simultaneous  occurrence  of  the 
assumed  system  of  residuals  is  then  (Art.  27) 

Y  =  i/i?/2  •••  pn  =  c^g-^'^('"^+''^+  -  +V).  (29) 

Now  if  m,  and  consequently  the  residuals  p,  are  to  be  so 
chosen  that  the  resulting  distribution  is  the  most  probable 
one  in  accordance  with  the  law  of  error,  these  quantities 
must  be  given  such  values  that  the  probability  Y  is  as 
great  as  possible.  But  this  will  be  secured,  evidently, 
by  making  Pi^  +  Pi"  +  •••  +  /°n^  as  small  as  possible,  as 
will  be  seen  at  once  from  (29) .  That  is  to  say,  m  should  be 
so  chosen  that  2/3^  is  a  minimum,  which  is  the  principle  of 
least  squares  stated  above. 

In  order  to  find  what  this  required  value  of  m  is,  we  may 

write 

^p^={s^-my  +  {s^-my+  ...  -V{Sn-my 

_._  =a  minimum. 

Hence 

-^2f)2  =  _2[(5i-m)  +  (^2-m)+  ...  +(5„-m)]=0, 
dm 

or  reducing,         m  =  '^i+'^2+  -  +^n ^  (3Q) 

n 

which  is  simply  the  arithmetical  mean  of  the  observations  s. 


THE  ERROR  EQUATION  61 

EXERCISES 

30.  1.  Show  how,  in  the  first  experimental  exercise  of 
Art.  11,  the  errors  of  aim  may  be  due  to  many  minor 
causes,  enumerating  as  many  such  possible  causes  as 
you  can  think  of. 

2.  Find  the  algebraic  sum  of  the  errors  of  measurement 
in  the  third  exercise  of  Art.  11 ;  also  the  algebraic  sum 
of  the  residuals. 

3.  Plot  the  curve  y  =  ce~~^^^\  giving  the  value  unity  to 
each  of  the  constants  c  and  L  This  may  be  done  by  use 
of  logarithms  {e  =  2.718  •••)•  Let  the  unit  abscissa  be 
10  squares  and  the  unit  ordinate  50  squares.  Compare 
with  the  error  curves  obtained  from  Exercises  1,  2  and  3 
of  Art.  11,  and  with  the  coin  problem  curve  obtained  in 
Art.  22. 

4.  Draw  a  smooth,  symmetrical  curve  which  follows 
as  closely  as  possible  the  irregular  curve  obtained  in 
Ex.  3,  Art.  11,  making  it  conform  to  the  known  prop- 
erties of  the  law  of  error  as  represented  in  Fig.  5.  From 
this  curve,  determine  the  relative  probabilities  of  the 
errors  of  magnitude  0°.l,  0°.2,  etc.,  out  to  5°.  By  locating 
the  points  of  inflection,  find  an  approximate  numerical 
value  for  h, 

5.  Plot  the  curve  represented  by 

2/  =  (2  -  x)2+  (3  -  xy  +  (4  -  xy  +  (5  -  xY  +  (6  -  xy. 
Has  it  a  minimum  point  ?     What  does  this  illustrate  ? 


62     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

6.  The  number  of  rays  in  the  lower  valve  of  a  certain 
species  of  Atlantic  mollusk  was  counted  in  508  individual 

cases.     Of  these, 

1  had  14  rays, 

8  had  15  rays, 

63  had  16  rays, 
154  had  17  rays, 
164  had  18  rays, 

96  had  19  rays, 

20  had  20  rays, 

2  had  21  rays. 

Plot  a  curve  in  which  abscissas  represent  the  number  of  rays 
and  ordinates  the  corresponding  number  of  individuals. 

What  is  the  probability  that  two  of  these  mollusks, 
picked  up  at  random,  will  each  have  exactly  fifteen  rays? 
(Data  from  Davenport,  Statistical  Methods.) 

7.  Tests  were  made  on  fifty  schoolboys  of  equal  age  to 
ascertain  strength  of  grip.  The  following  data  (Whipple, 
Manual  of  Mental  and  Physical  Tests)  are  in  hundreds  of 
grams. 


158 

210 

248 

296 

348 

175 

220 

262 

301 

350 

193 

225 

262 

310 

353 

197 

225 

267 

313 

375 

197 

225 

269 

315 

375 

200 

226 

270 

320 

403 

205 

235 

273 

323 

430 

206 

244 

280 

325 

440 

208 

244 

290 

330 

440 

210 

245 

294 

346 

508 

THE  ERROR  EQUATION  63 

Arrange  a  suitable  curve  showing  departures  from  the 
average  or  normal  strength  from  these  data. 

8.  About  two  hundred  individuals  were  tested  at  the 
University  of  Iowa  for  accuracy  of  tone  perception,  the 
results  being  expressed  by  the  number  of  vibrations  in 
the  departure  from  the  true  tone  (international  A,  435 
per  sec.)  that  the  individual  could  distinguish.  The  data 
are  expressed  in  per  cent. 

Departure,  Vib.  Per  Cent. 

1  13.8 

2  24.0 

3  25.5 
5  17.3 
8  7.3 

12  3.2 

17  1.6 

23  2.7 

30  or  over  4.6 

Plot  a  curve  representing  this  distribution,  and  discuss  its 
form. 

9.  Out  of  a  class  of  exactly  100  college  freshmen,  the  age 

1  was  16,  2  was  22, 

12  was  17,  1  was  23, 

31  was  18,  0  was  24, 

22  was  19,  0  was  25, 

18  was  20,  1  was  26. 
12  was  21, 

Plot  curve  and  discuss  its  form. 


64  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

10.   Out  of  over  100,000  public  school  grades  examined 
by  Mr.  L.  L.  Fishwild, 

1  per  cent,  were  50, 

1  per  cent,  were  55, 

2  per  cent,  were  60, 
2  per  cent,  were  65, 

5  per  cent,  were  70, 

6  per  cent,  were  75, 
13  per  cent,  were  80, 
13  per  cent,  were  85, 
25  per  cent,  were  90, 
23  per  cent,  were  95, 

9  per  cent,  were  100. 

Plot  curve  and  discuss  its  form. 


CHAPTER  V 

ON  THE  ADJUSTMENT  OF  INDIRECT 
OBSERVATIONS 

31.  Observations  on  Functions  of  a  Single  Quantity.  — 
It  has  been  pointed  out  that  measurements  are  seldom 
made  directly  upon  the  quantities  whose  values  are 
sought,  but  are  usually  made  upon  functions  of  them,  or 
functions  involving  them  with  other  unknown  quan- 
tities. The  former  case  being  the  simpler,  we  shall 
consider  it  first. 

As  a  specific  problem,  let  a  number  of  measurements 
be  made  upon  the  diameter  of  a  circle,  with  the  object  of 
determining  its  area.  That  is,  the  quantity  really  sought 
is  the  area,  but  the  direct  measurements  are  made  upon  the 
diameter,  a  function  of  the  area.  Supposing  the  observa- 
tions to  be  all  made  in  the  same  manner,  the  question 
arises,  what  is  the  most  probable  value  of  the  area  ?  Is  it 
the  arithmetical  mean  of  the  areas  computed  from  the 
separate  measurements  on  the  diameter,  or  is  it  the  area 
determined  by  taking,  as  the  diameter,  the  mean  of  the 
measurements  upon  it?  The  two  are  of  course  not  the 
same. 

This  question  may  be  answered  by  the  following  general 
deduction.  The  quantity  whose  most  probable  value  is 
F  65 


66     THEORY   OF   ERRORS   AND   LEAST   SQUARES 


sought  being  q,  and  the  function  of  it,  upon  which  the  ob- 
servations s  are  directly  made,  being /(g),  there  arise  the 
following  approximate  statements,  known  as  observation 
equations,  each  of  which  represents  one  of  the  n  measure- 
ments : 


fiq)  =  si, 
fiq)  =  S2, 

f{q)  =  Sn. 


(31) 


Si,  S2,  ••*,  Sn  are  the  results  of  readings  on  some  sort  of 
scale  or  measuring  instrument  applied  to  the  function 
directly  measured. 

The  errors  of  the  observations  are  represented  by 
Si—f{q),  etc.,  but  are  not  determinate.  It  is  the  most 
probable  value  of  q  that  we  are  seeking,  and  if  this  be  repre- 
sented by  m,  the  residuals  of  the  n  observations  are 


Pi  =  si  -f{m), 
pi  =  S2  -/(m), 

Pn  =Sn-  f{m). 


(32) 


There  is  no  reason  why  the  principle  of  least  squares 
should  not  apply  to  this  case  as  well  as  to  the  case  of  direct 
measurements,  since  the  law  of  error  distribution,  or  the 
"  law  of  departures,"  is  universal  in  its  scope.  As  relating 
to  this  sort  of  measurements,  then,  the  principle  of  least 
squares  takes  the  following  form :  The  most  probable 
value  of  an  unknown  quantity  that  can  be  derived  from  a 
set  of  observations  upon  one  of  its  functions  is  that  for  which 


ADJUSTMENT   OF   OBSERVATIONS  67 

the  sum  of  the  squares  of  the  residuals  arising  from  these 
observations  is  a  minimum. 

The  sum  above  referred  to  is  expressed  by 

^P'=[si-f(mW+[s2-fim)f+  '"  +K-/(m)P,  (33) 

in  which  m  may  be  regarded  as  a  variable  whose  value  is 
to  be  so  adjusted  as  to  render  2/3^  a  minimum.  This  con- 
dition requires  that 

dm 

or,  differentiating  (33), 

-  2[25-n/(m)]-^=0,     • 
dm 

/(m)=-'.  (34) 

n 

Therefore  m,  the  most  probable  value  of  q,  is  that  value 
whose  /-function  is  the  mean  of  the  observations  upon 

Thus,  the  most  probable  value  of  the  area  of  a  circle, 
as  determined  from  measurements  upon  the  diameter, 

is  -  times  the  square  of  the  arithmetical  mean  of  the  results 
4 

of  those  measurements.     A  multitude  of  other  illustrations 

of  this  principle  will  occur  to  any  one  familiar  with  such 

work. 

32.  Observation  Equations  for  More  Than  One  Un- 
known Quantity.  —  Very  frequently,  in  an  experimental 
research,  occasion  arises  to  determine,  not  merely  one,  but 


68     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

several,  unknown  quantities  or  constants  which  are  so 
involved  with  each  other  and  with  the  phenomena  directly 
observed  as  to  render  their  separate  measurement  im- 
possible.    The  following  illustrations  will  make  this  clear. 

In  the  use  of  the  zenith  telescope  for  finding  the  latitude 
of  a  station,  the  quantities  first  sought  are  the  zenith  dis- 
tances of  two  stars  selected  for  the  purpose.  The  sum  of 
the  zenith  distances  is  equal  to  their  difference  in  declina- 
tion, as  given  in  the  star  catalogues,  and  therefore  depends 
upon  the  results  of  many  very  precise  measurements 
made  with  other  instruments  at  fixed  observatories. 
The  difference  of  the  zenith  distances  is  measured  by  means 
of  the  micrometer  belonging  to  the  zenith  telescope, 
as  the  instrument  is  rotated  from  north  to  south  about 
the  vertical.  In  this  way,  neither  zenith  distance  is 
separately  determined,  both  being  found  by  the  simulta- 
neous solution  of  the  equations  arising  from  the  above 
observations. 

Again,  it  is  desired  to  find  the  relative  proportions  of 
sodium  chloride  (NaCl)  and  potassium  chloride  (KCl) 
in  a  mixture  of  the  two  salts.  Or  specifically,  in  a  given 
specimen  of  the  dry  mixture,  to  find  the  number  of  grams, 
X,  of  sodium  chloride  and  the  number,  y,  of  potassium 
chloride.     First  let  the  sample  be  weighed,  with  the  result 

Si.    Then  , 

X  -]-y  =  si. 

The  sample  is  now  dissolved  and  the  chlorine  precipitated 
with  silver  nitrate  (AgNOs),  and  the  total  amount  of 
chlorine  present  calculated  by  weighing  the  precipitated 


ADJUSTMENT   OF   OBSERVATIONS  69 

silver  chloride  (AgCl).  Denote  the  total  chlorine  by  ^2. 
Now,  sodium  chloride  is  0.6123  chlorine,  and  potassium 
chloride  is  0.4754  chlorine.  Hence  in  x  grams  of  sodium 
chloride  and  y  grams  of  potassium  chloride,  the  total 
chlorine  is  o.6123  x  +  0.4754  y  =  ^2, 

which  furnishes  the  second  observation  equation  necessary 
for  obtaining  x  and  y.  This  is  another  instance  in  which 
neither  of  the  unknown  quantities  is  measured  separately. 
Quite  often  only  certain  ones  of  the  unknown  quantities 
are  really  desired,  the  others  being  merely  troublesome 
corrections  or  instrumental  constants  which  must  be  de- 
termined or  eliminated.  The  method  of  procedure,  how- 
ever, is  the  same  in  this  as  in  any  other  case. 

33.  More  Observations  than  Quantities.  Normal 
Equations.  —  In  the  illustrations  of  the  preceding  article 
there  were,  in  each  case,  two  unknowns,  and  two  inde- 
pendent observations  were  necessary  to  determine  them. 
By  independent  observations  are  meant  observations 
made  on  a  different  principle,  or  under  such  different  con- 
ditions that  the  resulting  observation  equations  will  have 
different  coefficients  and  not  merely  different  absolute 
terms.  To  repeat  the  process  of  measuring  the  sum  of 
two  unknowns,  without  attempting  to  find  some  other 
relation  between  them  (as,  for  example,  their  difference 
or  their  product),  would  give  no  information  as  to  the 
separate  values  of  the  unknowns.  And,  in  general,  the  de- 
termination of  /  unknown  quantities  requires  a  knowledge 
of  /  independent  and  consistent  relations  between  them. 


70  THEORY   OF   ERRORS   AND   LEAST   SQUARES 


If  measurements  could  be  made  without  error,  the  solu- 
tion of  the  /  independent  observation  equations  formed 
from  such  measurements  would  give  us  the  values  of  the  / 
unknowns  exactly;  more  than  /  measurements  would  be 
superfluous.  But,  as  in  the  simpler  case  of  a  single  un- 
known, the  existence  of  accidental  errors  makes  it  desir- 
able to  get  as  many  observations  as  possible,  and  to 
devise  some  means  of  averaging  them  so  as  to  find  the 
most  probable  value  of  each  of  the  unknowns.  This  prob- 
lem is  the  most  important  that  arises  in  least  squares. 

Let  there  be  n  observations  upon  functions  of  the  I  un- 
known connected  quantities  qi,  q2,"',  Qi  {n>l),  and  let 
the  series  of  resulting  observation  equations  be  repre- 
sented by 


h  (qu  qi, 


qi)  =  si, 
qi)  =  ^2, 


fn{qi,  ?2,    •••,   qi)  =Sn. 


(35) 


Here,  as  in  the  simpler  cases,  there  are  errors  and  residuals 

obeying  the  same  law  of  error  distribution  set  forth  in 

the  error  equation.     We  are  seeking  to  obtain  the  most 

probable  values,  mi,   m2,  •••,  nii,  of  the   unknown    (and 

unknowable)    quantities    q    that    the    observations    will 

furnish,  and  when  these  are  found,  the  n  residuals  will  be 

given  by 

Pi  =  si-fi  (mi,  m2,  •••,  TUi), 

P2  =  S2-f2  {rni,  rui,  •••,  m,),  ^ 


P«  =  Sn-fn('rrii,m2,  •-,  m,). 


ADJUSTMENT   OF   OBSERVATIONS 


71 


The  principle  of  least  squares  may  now  be  slightly  modi- 
fied in  wording  to  fit  this  case,  thus :  The  most  probable 
values  of  unknown  quantities  connected  by  observation  equa- 
tions are  those  which  will  render  the  sum  of  the  squares 
of  the  residuals  arising  from  the  observation  equations  a 


minimum. 


It  is  possible,  through  an  application  of  this  principle, 
to  reduce  the  n  residual  equations  (36)  to  a  number  /, 
equal  to  the  number  of  unknowns,  which  can  then  be 
solved  for  the  most  probable  values  m.  The  process 
may  be  regarded  as  finding  from  the  n  observation  equa- 
tions (35)  a  set  of  /  most  probable  equations  whose  solution 
will  give  the  most  probable  values  of  the  unknowns  q. 

From  the  principle  of  least  squares,  the  sum  pi^  +  p2^  + 
•••  +Pn^  must  be  a  minimum,  and  in  order  that  mi,  ?7i2, 
•••,  mi  may  be  so  selected  that  this  will  be  the  case,  the 
first  partial  derivative  of  this  S/a^  with  respect  to  each 
of  those  quantities  must  be  zero.  (See  any  calculus.) 
That  is, 

_d_ 

dmi 


^[s-f  (mi,  m2,  •••,  mi)Y  =  0, 


dm2 


^mi    1 


(37) 


The  equations  (37)  resulting  from  these  differentiations 
are  the  most  probable  or  normal  equations  required,  and 


72  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

being  /  in  number,  will  yield,  on  solution,  the  most  probable 
values  mi,  mz,  •••,  mi,  which  are  sought. 

Equation  (34)  is  a  normal  equation,  containing  only 
one  unknown,  m. 


34.  Reduction  of  Observation  Equations  of  the  First 
Degree.  —  In  nearly  all  cases  in  which  the  method  of  least 
squares  is  used  in  the  reduction  of  observations  in  accord- 
ance with  the  foregoing  theory,  the  observation  equations 
are  either  all  cf  the  first  degree,  or  they  may,  by  suitable 
substitutions,  be  replaced  by  equivalent  observation 
equations  which  are  of  the  first  degree.  The  mathemati- 
cal operations  required  in  finding  the  normal  equations 
are  then  comparatively  simple,  and  can  be  performed  with- 
out any  knowledge  of  calculus. 

Let  the  n  first  degree  observation  equations  upon  the  I 
quantities  q  (corresponding  to  (35))  be  as  follows : 


(38) 


■  tti^i  +  biq2  +  Ciqs  + h  nqi  =  Si, 

a^qi  +  hqi  +  ^273  H h  r^qi  =  s^, 

anqi  +  M2  +  ^n^sH h  Tnqi  =  Sn. 

The  residuals  will  then  be 

Pi  =  si-  {aiini  +  biJUi  +  •••  +  nmi), 

P2  =  S2  —  {a^mi  +  627772  +  •  •  •  +  r2mi 

), 
). 

Pn  =  Sn-  {cinrni  +  Knii  +  •••  +  rnTTii 

(39) 


Only  one  term  in  each  of  these  expressions  contains  mi ; 
denote  the  balance  of  the  expression  in  each  case  by  a  single 


ADJUSTMENT   OF   OBSERVATIONS  73 

letter,  as  B.     Then 

pi  =  —  airrii  +  Bi,  etc., 
and 


Z/)2  =  ( -  aivii  +  BiY  +•••+(-  a„mi  +  B^Y. 
with  respect  to  mi,  as  pe: 
2/^2  =  —  2  tti  ( —  aimi  +  Bi) 


Differentiating  with  respect  to  mi,  as  per  equations  (37), 
d 
drrii 


-2an{-  anTThi  +  Bn)  =  0. 

Or  dividing  by  2  and  remembering  that  —  am  +  B  =  p 
in  each  term, 

-  aipi  -  aiPi anpn  =  0.  (40) 

This  is  the  normal  equation  pertaining  to  mi,  and  corre- 
sponds to  the  first  of  equations  (37) . 

This  result  may  be  directly  obtained  by  multiplying 
each  of  the  residuals  (39)  by  the  coefficient  of  mi  in  the  ex- 
pression for  that  residual,  adding  the  results  and  equating 
the  sum  to  zero. 

The  remainder  of  the  I  normal  equations  required  are 
determined  with  respect  to  m2,  ms,  •••,  m^  in  the  same 
manner. 

The  foregoing  processes  may  be  summed  up  in  the 
following  rule :  To  adjust  a  set  of  observation  equations  of 
the  first  degree,  write  the  expression  for  the  residual  corre- 
sponding to  each  observation  equation,  multiply  it  by  the 
coefficient  of  the  first  unknown,  in  that  expression,  add  the 
products  and  equate  their  sum  to  zero.  The  result  is  the 
normal  equation  pertaining  to  the  said  first  unknoivn. 
Do  likewise  for  each  of  the  other  unknowns.     Then  solve 


74     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

the  I  normal  equations  thus  formed  for  the  desired  most 
probable  values,  mi,  m2,  "',mi. 

Let  the  student  prove  that  taking  the  arithmetical  mean 
of  a  number  of  direct  observations  upon  a  single  quantity 
is  merely  a  special  application  of  this  rule. 

35.  Illustrations  from  Physics.  —  It  will  be  of  material 
assistance  to  the  student  to  have  presented  at  this  point 
a  number  of  actual  examples  illustrating  the  application 
of  least  square  adjustment  in  various  departments  of 
exact  science.  These  examples  are  not  "  made  up  "  for 
the  purpose ;  they  are  drawn  from  actual  experimental 
notes  on  research  or  field  work. 

1.  Bridge  Wire.  —  It  was  desired  to  measure  the  total 
resistance  of  a  Wheatstone  bridge  wire  and  at  the  same 
time  to  calibrate  it,  by  comparison  with  a  standardized 
bridge  of  another  type.  The  unknown  (and  unessential) 
resistance  of  the  connections  had  also  to  be  reckoned  with 
and  eliminated.  The  wire  was  100  cm.  long,  and  the  meas- 
urement was  conducted  by  observing  the  resistance  of  the 
first  10  cm.,  then  of  the  first  20  cm.,  etc.,  and  finally  of  the 
whole  wire,  the  connections  entering  each  time  as  a  con- 
stant term  in  the  observed  resistance.     The  results  follow. 


No.  Cm. 

Resist., 

No.  Cm. 

Resist., 

Measured 

Ohms 

Measured 

Ohms 

10 

0.116 

60 

0.595 

20 

0.205 

70 

0.675 

30 

0.295 

80 

0.760 

40 

0.388 

90 

0.850 

50 

0.503 

100 

0.926 

ADJUSTMENT   OF   OBSERVATIONS  75 

Let  X  be  the  total  resistance  of  the  bridge  wire,  and  c  that 
of  the  connections.  These  are  the  two  unknowns,  the 
first  of  which  is  to  be  obtained  with  all  possible  precision, 
the  second  to  be  eliminated,  as  a  mere  correction.  Mathe- 
matically they  are  equally  important.     The  observation 

equations  are  ^  ^       ,  r\  tin 

^  0.1  ar  +  c  =  0.116, 

0.2x-{-c  =  0.205, 

O.Sx  +  c  =  0.295, 

0.4  X  +  c  =  0.388, 

0.5  a:  +  c  =  0.503, 

0.6X  +  C  =  0.595, 

0.7X  +  C  =  0.675, 

0.8  X  +  c  =  0.760, 

0.9X  +  C  =  0.850, 

1.0  a: +  c  =  0.926. 

In  practice  we  need  not  take  the  trouble  to  change 
symbols  in  distinguishing  between  the  true  and  most  prob- 
able values  of  the  unknown  ("  q  "  and  "  m  ").  If  x  and  c 
now  represent  the  most  probable  values  sought,  the  first 
residual  is  pi  =0.116  —  (0.1  a:  +  c),  etc.  Let  the  student 
apply  the  rule  developed  in  the  preceding  article  to  obtain 
the  two  normal  equations,  which  he  will  find  to  be 

3.85  a:  +  5.5  c  =  3.686, 
5.5  a;  +  10  c  =  5.313, 

the  solution  of  which  gives 

X  =  0.926  ohms, 
c  =  0.022  ohms. 


76     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

Let  the  student  select  any  two  of  the  observation  equa- 
tions and  solve  them  for  x  and  c,  comparing  the  results 
with  these  most  probable  ones.     The  accompanying  figure 


—•90 

—eo 

—  70 
— «0 
-50 

10 

^ 

h 

-s 

^ 

Y 

K 

Y 

^ 

l^ 

^ 

^ 

—JO 

^ 

^ 

^ 

—.10 

^ 

^ 

^ 

^ 

h  cms. 

" 

lO 

1 

20 

1 

30 

1 

40 
1 

50 

1 

60 

1 

TO 

1 

80 

1 

r 

100 

1 

. ..._    J 

Fig.  6 

shows  the  plotted  observations,  together  with  the  straight 
line  I 


0.926^^  +  0.022 
100 


R> 


upon  which  they  all  should  lie  were  there  no  errors  in  the 
measurements  nor  irregularities  in  the  wire  itself.  The 
departures  of  the  plotted  points  from  this  most  probable 
line  represent  the  residuals  of  the  ten  observations. 

2.  Balance  Constants. — The  general  theory  of  the  equal- 
arm  balance  is  somewhat  complicated,  but  in  the  equation 
used  to  express  the  sensibility  in  terms  of  the  load,  the 
various  instrumental  constants  may  all  be  involved  in 
two  quantities  a  and  6,  the  equation  being 

a  -{-bw  =  -' 
s 


ADJUSTMENT   OF   OBSERVATIONS 


77 


Here  iv  is  the  load  on  either  pan  (grams)  and  s  the  gram 
sensibiHty,  or  one  thousand  times  the  deflection  produced 
by  a  milligram  weight  laid  on  one  pan.  The  constants  a 
and  b  are  to  be  estimated  from  the  following  observations. 


w 

s 

w 

s 

Grams 

Scale  Div. 

Grams 

Scale  Div. 

0 

2212 

50 

2389 

10 

2265 

75 

2449 

20 

2320 

100 

2563 

30 

2343 

125 

2590 

40 

2316 

The  observation  equations  are  then 

a  +  0  6  =  1  H-  2212, 
a+  10  6  =  1  --  2265, 
a  +  20  6  =  1  -^  2320, 
a  +  30  6  =  1  -  2343, 
a  +  40  6  =  1  ^  2316, 
a  +  50  6  =  1  ^  2389, 
a  +  75  6  =  1  -  2449, 
a  +  100  6  =  1  -v-  2563, 
a  +  125  6  =  1  -  2590. 

The  adjustment  of  these  by  the  foregoing  method  gives  as 
the  most  probable  values  sought, 

a  =  -\-  0.0004466, 
h  =  -  0.000000518. 


Let  the  student  perform  this  reduction. 


78  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

36.   Illustrations  from  Chemistry. 

1.  Volumetric  Solutions.  —  It  is  desired  to  test  certain 
acid  and  alkaline  solutions  to  be  used  in  volumetric 
chemical  analysis,  in  order  to  ascertain  their  exact  strengths. 
Two  common  reagents,  in  the  form  of  tenth-normal  solu- 
tions, may  be  tested  first,  then  others  may  be  compared  to 
these.  If  the  two  reagents  chosen  be  hydrochloric  acid 
and  potassium  hydroxide,  the  following  procedure  may 
be  employed. 

A  quantity  of  each  solution  is  placed  in  an  accurately 
graduated  burette,  the  two  burettes  being  supported 
side  by  side.  A  small  amount  (say  about  0.2  g.)  of  finely 
pulverized  pure  calcium  carbonate  (chalk,  CaCOs)  is 
carefully  weighed,  placed  in  a  white  porcelain  dish  and 
treated  with  an  excess  (say  about  50  cc.)  of  the  HCl  solu- 
tion from  the  burette,  the  amount  being  accurately  ob- 
served. The  chalk  dissolves  and  neutralizes  part  of  the 
acid,  the  CO2  gas  escaping.  The  porcelain  dish  is  now 
set  under  the  KOH  burette,  and  just  enough  of  the  alka- 
line solution  allowed  to  flow  into  it  to  render  it  exactly 
neutral,  this  point  being  determined  by  a  drop  or  two  of 
methyl  orange  or  other  sensitive  indicator  previously  added 
to  the  mixture  in  the  dish.  The  amount  of  KOH  solution 
thus  used  is  also  carefully  noted.  Part  of  the  acid  is 
neutralized  by  the  CaCOa  and  the  remainder  by  the  KOH. 
The  chemical  equations  representing  the  two  reactions 
are  as  follows: 

72.36  99.32 

(I)  2  HCl  +  CaCOa  =  CaCl2  +  CO2  +  H2O, 


ADJUSTMENT   OF   OBSERVATIONS  79 

36.18         55.70 

(II)      HCl  +  KOH  =  KCl  +  H2O. 

The  small  numbers  above  the  symbols  are  obtained  from 
the  molecular  weights,  and  represent  the  relative  weights 
of  the  substances  engaging  in  the  reaction. 


Unknown. 


Let  gi  =  wt.  HCl  in  1  cc.  HCl  sol. 

^2  =  wt.  KOH  in  1  cc.  KOH  sol. 

a  —  total  volume  HCl  sol.  used. 

a  =  vol.  HCl  sol.  neutralized  by  CaCOs. 
a  —  a  =  vol.  HCl  sol.  neutralized  by  KOH. 

h  =  vol.  KOH  sol.  used  in  neutraUzation. 

c  =  wt.  CaCOa  powder  used. 

Then  aqi  =  wt.  HCl  neutralized  by  CaCOa. 

(a  —  a)qi  =  wt.  HCl  neutralized  by  KOH. 

bq2  =  wt.  KOH  used. 
From  (I) 

aqi'.c  -=  72.36 :  99.32  =  0.73, 

or  aqi  =  0.73  c. 

From  (II) 

(a  -  a)qi :  6^2  =  36.18  :  55.70  =  0.65, 

or  aqi  —  aqi=  0.65  bq2. 

From  these  two  equations  a  is  eliminated  by  addition, 

giving  finally 

aqi  -  0.65  6^2  =  0.73  c. 


80     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

This  is  an  observation  equation,  the  quantities  a,  h,  c 
having  been  measured,  and  gi,  g2  being  the  two  unknowns ; 
and  a  series  of  such  experiments  (at  least  two)  will  yield 
the  most  probable  values  required.  In  some  of  the  ex- 
periments the  CaCOs  powder  may  be  omitted  entirely, 
giving  c  =  0  ;   but  not  in  all  of  them.     (Why  ?)    • 


a 

6 

c 

Vol.  HCI 

Vol.  KOH 

Wt.  CaCOa 

Sol.  used 

Sol.  used 

Powder  used 

cc. 

cc. 

g- 

50.00 

10.33 

0.1779 

50.00 

7.88 

0.1936 

11.23 

9.98 

none 

11.25 

10.00 

none 

11.25 

10.00 

none 

11.34 

10.10 

none 

The  above  data  yield  the  following  observation  equa- 
tions : 

50  gi  -  0.65  X  10.33  g2  =  0.73  X  0.1779, 

50  qi  -  0.65  X  7.88  ^2  =  0.73  X  0.1936, 

11.23  gi  -  0.65  X  9.98  g2  =  0, 

11.25  gi  -  0.65  X  10.00  g2  =  0, 

•       11.25  gi  -  0.65  X  10.00  g2  =  0, 

11.34  gi  -0.65  X  10.10  g2  =  0. 

Let  the  student  reduce  these  to  normal  equations  and  solve 
for  the  most  probable  values  of  gi  and  g2. 

2.    Pyknometer  Constants.  —  The  expansion  of  a  pyk- 
nometer  (specific  gravity  bottle),  like  any  solid,  is  in  ap- 


ADJUSTMENT   OF   OBSERVATIONS  81 

proximate  accordance  with  the  linear  law 

V  =  Vo  +  Kt, 

V  being  the  capacity  at  temperature  t,  Vq  the  capacity  at 
zero  and  K  a  constant  involving  the  coefficient  of  expan- 
sion of  the  glass.  The  two  constants  Vq  and  K  must  be 
experimentally  determined  from  time  to  time  for  any 
pyknometer  that  is  used  in  accurate  measurements  of 
density.  This  may  be  done  by  finding  the  capacity  at 
several  different  temperatures  over  the  required  range.* 
The  following  is  a  tabulation  of  eight  such  determinations, 
using  distilled  water  and  corrected  for  buoyancy  of  the  air. 


t 

V 

t 

V 

19°.20 
19.75 
25.61 
30.92 

25.2628  cc. 
.2634 
.2664 
.2681 

35°.50 
39.30 
39.75 
^  46.45 

25.2687  cc. 
.2691 
.2692 
.2734 

Let  the  student  form  the  eight  observation  equations  and 
the  two  normal  equations,  and  reduce  for  the  most  prob- 
able values  of  Vq  and  K.  (The  approximate  answers  are, 
T^o  =  25.2509,  K  =  0.0005244.) 

37.   Illustrations  from  Surveying. 

1.  Locating  a  Distant  Station.  —  Some  of  the  best 
writers  on  surveying  strongly  recommend  the  use  of  rec- 

*  If  the  range  be  large,  K  will  vary  somewhat.  The  range  may  be 
subdivided,  say  into  lO-degree  intervals,  and  the  constants  found  for 
each ;  or  better,  a  quadratic  relation  assumed,  with  three  constants.  See 
Art.  45. 


82     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

tangular  coordinates  in  surveying  and  mapping ;  certainly 
their  use  reduces  many  calculations  to  a  more  scientific 
basis.     The  problem  in  hand  is  as  follows :   Given,  the 


Fig.  7 


coordinates  of  a  number  of  stations  A^  B,  C,  etc.,  with 
reference  to  an  origin  0,  and  the  bearing  of  an  unknown 
station  P  from  each  of  these  stations;  to  find  the  most 
probable  coordinates  of  P.     For  instance,  the  unknown 


ADJUSTMENT   OF   OBSERVATIONS 


83 


station  P  is  a  reef  near  the  coast  along  which  the  points 
A,  B,  etc.,  are  located.  The  numerical  data  are  as  follows, 
for  five  stations. 


Station 

Coordinates  (Ft.) 

Bearing  of  P 

Vector 

Angle  B 

E. 

N. 

A 

1785 

1501 

S.    58°  57'  W. 

PA 

3r  3' 

B 

1372 

2020 

S.    22       5     W. 

PB 

67     55 

C 

1052 

1971 

S.      5     29     W. 

PC 

84    31 

D 

909 

1609 

S.      4     43     E. 

PD 

94     43 

E 

620 

1533 

S.    32     43     E. 

PE 

122     43 

The  vectorial  angles  in  the  last  column  are  the  angles 
made  by  the  vectors  PA,  PB,  etc.,  with  the  line  drawn 
eastward  through  P,  calculated  from  the  given  bearings. 
Using  coordinates  x  and  y  to  locate  P,  and  Xa,  ya,  etc.,  for 
A,  etc.,  we  can  write 


Xa         X 


=  tan  dn 


or 


X  tan  6a  —  y  =  Xa  tan  da  —  yc 


that  is. 


X  tan  31°  S'  -y  =  1785  tan  31°  3'  -  1501, 

etc.,  as  the  observation  equations,  there  being  as  many  of 
these  as  there  are  known  stations.  These  equations,  being 
of  the  first  degree  in  x  and  y,  may  be  adjusted  in  the  usual 
manner.  Let  the  student  do  this.  (The  results  should 
be,  approximately,  x  =  930,  y  =  1000 ;  that  is,  P  =  930 
E.,  1000  N.) 


84     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

2.    Relative  Levels  of  Stations.  —  The  next  illustration 

is  taken  from  Merriman's  Least  Squares,  and  is  typical  of 

many  kinds  of  measurements  in  which  the  quantity  sought 

is  measured  by  parts  or  segments.     The  same  method  is, 

for  example,  applied  to  a  number  of  angles  at  one  station. 

Given,  a  number  of  determinations  on  the  relative  altitudes 

of  several  stations,  obtained  by  precise  leveling,    to  find 

the  most  probable  values  of  their  altitudes  above  one  of 

them  taken  as  a  datum.     Following  are  the  results  of  the 

levelings. 

A  above  0  573.08  ft. 

B  above  A     2.60  ft. 

B  above  0  575.27  ft.   • 

C  above  B  167.33  ft. 

D  above  C      3.80  ft. 

D  above  B  170.28  ft. 

I)  above  E  425.00  ft. 

E  above  0  319.91  ft.  (one  way) 

E  above  0  319.75  ft.  (another  way) 

Representing  by  a,  6,  etc.,  the  elevations  of  the  respective 
stations  above  0  as  a  datum,  the  following  simple  observa- 
tion equations  at  once  result. 

a  =  573.08 
h-a  =  2.60 

h  =  575.27 
c-h  =  167.33 
d-c  =  3.80 


ADJUSTMENT   OF   OBSERVATIONS  85 

d-b  =  170.28 
d-e  =  425.00 

e  =  319.91 

e  =  319.75 

The  student  can  readily  adjust  these  in  the  usual  manner. 
It  will  be  interesting  in  this  case  to  compare  the  adjusted 
values  of  a,  h  and  e  with  their  values  as  directly  measured. 

38.   Illustrations  from  Astronomy. 

1.  Errors  of  the  Transit  and  Clock. — Astronomical  time 
is  ascertained,  at  any  observatory,  by  observations  upon 
the  stars.  To  this  end  an  instrument  not  unlike  a  sur- 
veyor's transit  is  used.  It  is  larger,  however,  and  fixed 
on  a  solid  pier,  and  is  incapable  of  rotating  horizontally, 
being  swung  in  the  vertical  plane  of  the  meridian.  This 
instrument  is  the  astronomical  transit  or  the  meridian 
circle. 

When  used  for  time  observations,  the  telescope  is  set  at 
the  proper  angle  of  altitude  for  some  star  to  traverse  its 
field  as  it  crosses  the  meridian.  The  exact  sidereal  time 
of  meridian  passage,  or  transit,  is  known  as  the  right 
ascension"^  of  the  star, -and  is  given  in  the  star  catalogues. 
In  order  to  correct  the  clock,  therefore,  it  is  necessary  only 
to  note  at  what  time,  by  the  clock,  the  star  is  actually  ob- 
served to  cross  the  meridian. 

*  Right  ascension  on  the  celestial  sphere,  as  shown  by  the  star  maps, 
is  closely  analogous  to  longitude  on  the  earth,  only  it  is  usually  expressed 
in  hours,  minutes  and  seconds,  reading  toward  the  east.  Declination 
corresponds  to  terrestrial  latitude. 


86  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

On  account  of  the  extreme  accuracy  demanded  in  as- 
tronomical work,  this  apparently  simple  procedure  re- 
quires the  elimination  of  certain  recognized  instrumental 
errors.  These  are :  (1)  the  level  error,  arising  from  the 
non-horizontality  of  the  bearings  or  trunnions  on  which 
the  telescope  turns,  so  that  its  revolution  does  not  exactly 
coincide  with  the  meridian  plane ;  (2)  the  azimuth  error, 
or  failure  of  this  axis  of  rotation  to  coincide  with  the  east 
and  west  line,  which  has  a  similar  effect  on  the  plane 
of  rotation ;  (3)  the  collimation  error,  due  to  the  fact  that 
the  cross-wires  in  the  telescope,  which  determine  its  line 
of  sight,  are  not  exactly  in  the  optic  axis,  being  a  little  to 
one  side  of  the  center  of  the  field.  In  addition  to  these, 
there  is  the  error  of  the  clock,  which  is  the  quantity  really 
wanted.  The  level  error  is  ascertained  by  a  direct  applica- 
tion of  the  stride  level  resting  on  the  trunnions  and  having 
a  very  sensitive  graduated  spirit-bubble.  The  other  errors 
must  be  found  simultaneously  from  several  observations 
on  different  stars,  the  level  error  reading  being  simply  a 
part  of  the  determination. 

Without  entering  into  the  applications  of  spherical  as- 
tronomy required,  it  may  be  simply  stated  that  the  ob- 
servation equations  involved  are  of  the  first  degree.     If 

^1  =  the  true  clock  error  (clock  minus  true  time), 
^2  =  the  azimuth  error, 
^3  =  the  collimation  error, 
/  =  the  level  error, 

all  being  expressed  in  seconds  of  time,  then  the  form  of 


ADJUSTMENT   OF   OBSERVATIONS 


87 


the  observation  equation  is 

Qi  +  aq2  +  cqz  =  d  —  bl, 

d  being  the  apparent  clock  error,  or  the  time  indicated  by 
the  clock  at  apparent  transit  minus  the  true  time  of  transit, 
or  right  ascension,  of  the  star.  The  quantities  a,  b  and  c 
are  known  as  Meyer's  coefficients,  and  are  calculated  from 
the  following  formulas : 

sin  (X— 5) 


a  = 


cos  5 


7  _  cos  (X— 5) 
cos  8 

c  =  sec  8, 

in  which  X  is  the  latitude  of  the  observatory  and  8  the  dec- 
lination of  the  star  used.  Tables  of  these  coefficients  are 
at  hand  in  every  observatory. 

Of  course  three  observations  on  different  stars,  at  least, 
are  required  to  determine  gi  and  eliminate  ^2  and  q^.  If 
more  are  made,  least-square  reduction  may  be  applied 
to  their  adjustment.  Following  is  a  typical  set  of  data 
of  this  sort,  based  on  the  observed  transits  of  six  stars. 


Iowa  City,  Iowa, 

Lat.  41°  40' 

November  16,  1896 

Star 

Declina- 
tion 

Right 
Ascension 

Observed 
Time  of 
Transit 

I 

a 

b 

c 

«  Draconis    .     . 
iSCeti      .     .     . 
y  Cassiopeiae     . 
0"  Ursae  Majoris 
f  Cygni    .     .     . 
a  Cephei       .     . 

109°  38' 

-18    33 

60      9 

112    27 
29   48 
62      9 

h,  m.    s. 

0  29    4.28 

0  38  26.50 

0  50  30.72 

21     1  21.85 

21     8  32.81 

21  16    6.35 

h.  m.     s. 

0  29  39.14 

0  39  34.07 

0  52    4.77 

21     2     1.41 

21     9  52.14 

21  17  42.64 

a. 
OAT 
0.44 
0.38 
0.59 
0.59 
0.59 

2.76 
0.91 

-0.64 
2.47 
0.24 

-0.75 

-1.12 
0.52 
1.91 

-0.86 
1.13 
2.01 

-2.97 
1.05 
2.01 

-2.61 
1.15 
2.14 

88      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

No  allowance  has  here  been  made  for  any  error  in  the 
clock's  rate  during  the  progress  of  the  observations. 

2.  Parallax  and  Proper  Motion.  —  Stellar  parallax  is  the 
apparent  change  in  the  position  of  a  star,  during  the  year, 
caused  by  the  earth's  motion  in  its  orbit.  In  addition  to 
this,  there  is  the  actual,  or ''  proper,"  motion  of  the  star 
itself  through  space.  These  two  are  superposed  and  pro- 
duce one  resultant  effect  upon  the  star's  apparent  posi- 
tion at  any  time.  Their  separation  into  distinguishable 
components  is  the  problem  here  presented. 

Modern  astronomical  measurements  are  conducted 
very  largely  by  photography.  The  star  in  question  is 
photographed  on  the  same  plate  with  others  so  immensely 
farther  away  that  they  have  no  perceptible  parallax  or 
proper  motion,  and  then  the  positions  of  the  images  are 
measured  at  leisure  on  very  accurate  measuring  machines. 

Let  TT  =  the  parallax  in  a  given  direction, 
ft  =  the  proper  motion  in  that  direction, 
s  =  the  measured  displacement  of  the  star  in  tHat 
direction,    with   reference   to    its    apparent 
position  at  some  previous  date  T  days  past. 

Then    the    observation    equation  is  shown  in  practical 

astronomy  to  be 

Pt  +  Tfi  -\-  c  =  s. 

P  is  the  parallax  factor,  easily  calculated  from  the  direc- 
tion of  the  star  and  the  position  of  the  earth  in  its  orbit. 
c  is  an  unknown  constant,  depending  on  the  f)eculiarities 
of  the  measuring  machine,  and  to  be  eliminated.     The 


ADJUSTMENT   OF   OBSERVATIONS  89 

three  unknowns  are,  then,  tt,  ft  and  c.  The  coefficients  P 
and  T  and  the  quantity  s  are  varied  by  making  observa- 
tions on  many  different  dates,  and  from  the  resulting 
series  of  observation  equations,  the  most  probable  values 
of  the  proper  motion  and  the  parallax  are  obtained.  The 
latter  gives  the  most  probable  distance  of  the  star.  The 
details  of  the  process  being  somewhat  technical,  no  numeri- 
cal example  is  here  given. 

39.  Observation  Equations  Not  of  First  Degree.  —  If 
the  observation  equations  are  not  of  the  first  degree,  re- 
course may  be  had  to  the  general  method  explained  in 
Art.  33,  that  is,  to  the  application  of  the  principle  of  least 
squares  through  the  general  equations  (37).  This  would 
often  lead,  however,  to  normal  equations  that  would  be 
exceedingly  inconvenient  to  solve. 

In  many  such  cases,  the  difficulty  may  be  at  once  avoided 
by  a  suitable  application  of  logarithms.  A  standard 
measurement  in  the  physical  laboratory,  for  example,  is 
the  simultaneous  determination  of  the  magnetic  field  of 
the  earth  H  and  the  magnetic  moment  M  of  the  bar 
magnet  used  for  the  purpose.  One  experiment  gives  the 
product,  ^^^^^^  ^^j^ 

and  another  the  quotient 

-^  =  S2,  (42) 

of  the  unknown  quantities.  These  observation  equations 
may  be  made  linear  by  using  instead  of  M  and  //,  as 


90        THEORY    OF    ERRORS    AND    LEAST   SQUARES 


(43) 


unknowns,  their  common  logarithms : 

log  M  -[-\ogH  =  log  si, 

log  M  —  log  //  =  log  52, 

the  most  probable  values  of  log  M  and  log  H  being  then 
found  in  the  usual  manner. 

Again,  the  solubility  of  a  chemical  salt  is  given  by  the 
theoretical  formula*  ^, 

s  =  s^^^^^\  (44) 

in  terms  of  the  centigrade  temperature  t.  Sq  is  the 
solubility  of  the  salt  at  0°  C.  and  c  is  a  constant  depend- 
ing on  its  heat  of  solution,  ^o  and  c  are  unknowns,  to  be 
determined  for  each  substance  by  means  of  several  meas- 
urements on  s  at  different  temperatures.  For  this  purpose 
the  observation  equation  may  be  written 

t 


log  So  + 


log  e  '  c  =  log  s, 


(45) 


273  +  f 

the  most  probable  values  of  c  and  log  ^o  being  the  values 
directly  found.  The  following  data  pertain  to  the  solu- 
bility of  potassium  chlorate  (KCIO3)  in  water. 


t 

8  (obs.) 

8  (calc.) 

O'' 

0.0247 

5 

0.0299 

.0317 

10 

.0406 

.0402 

15 

.0512 

.0507 

20 

.0672 

.0634 

25 

.0774 

.0787 

30 

.1027 

.0970 

35 

.1145 

.1187 

40 

.1405 

.1444 

*  See  Arrhenius,  Electrochemistry. 


ADJUSTMENT   OF   OBSERVATIONS  91 

The  eight  observations  on  t  and  s  furnish  eight  observation 
equations  of  the  above  form,  which  when  adjusted  give 
as  most  probable  values  log  ^o  =  —1.6073,  whence  ^o  = 
0.0247 ;  and  c  =  13.82.  The  solubility  formula  for  this 
substance  may  now  be  written  in  its  original  form,  or  more 
conveniently  retained  in  the  logarithmic  form : 

log  s  =  6.0014 ^ 1.6073, 

^  273  +  ^ 

from  which  the  values  of  s  given  in  the  third  column  are 
calculated.  The  student  will  find  it  instructive  to  plot 
the  observations  on  t  and  s,  and  also  the  smooth  curve 
corresponding  to  the  calculated  values  of  s.  It  would  be 
difficult  to  imagine  a  more  typical  application  of  least- 
square  adjustment  than  the  one  just  given. 

Another  method  of  procedure  when  the  observation 
equations  are  not  of  the  first  degree,  somewhat  analogous 
to  Horner's  method  of  approximation  for  higher  algebraic 
equations,  is  explained  in  Note  B,  Appendix. 

40.  Observations  upon  Quantities  Subject  to  Rigorous 
Conditions.  —  It  often  happens  that  unknown  quantities 
involved  in  observation  equations  are  further  connected 
by  known  mathematical  conditions,  which  the  final  ad- 
justed values  must  rigorously  satisfy.  For  example,  the 
most  probable  values  of  the  angles  of  a  triangle  could  not 
be  a  set  of  angles  whose  sum  is  other  than  exactly  180° ; 
the  sum  of  all  the  percentages  in  a  chemical  analysis 
must  be  100;  etc.  Observations  upon  such  quantities 
are  known  as  conditioned  observations. 


92        THEORY   OF   ERRORS   AND   LEAST   SQUARES 

Suppose  that  the  results  of  measurements  upon  the 
three  angles  of  a  triangle  are 


qi  =  ^2, 
qz  =  sz- 


(46) 


These  are  the  observation  equations.  To  this  list  there 
must  be  added  a  fourth  equation,  namely : 

91  +  ^2  +  gs  =  180°,  (47) 

which  is  called  an  equation  of  condition.  It  differs  from 
the  others  in  that  it  is  known  to  be  exactly  true,  while 
the  others  are  not.  The  three  most  probable  values, 
when  deduced,  must  satisfy  this  equation  exactly ;  the 
others  must  be  satisfied  as  nearly  as  may  be.  This  equa- 
tion of  condition  cannot,  therefore,  be  classed  as  an  ob- 
servation equation  and  treated  like  the  others. 

In  general,  we  may  have  n  observations  involving  /  un- 
knowns, which  are  further  subject  to  m  rigorous  conditions, 
expressed  as  equations  of  condition,  m  must  be  less  than  / ; 
for  if  equal  to  it,  the  unknowns  would  be  absolutely  de- 
termined by  the  given  conditions,  and  the  measurements 
would  be  superfluous ;  and  if  greater,  no  set  of  quantities 
could,  in  general,  be  found  to  satisfy  all  the  conditions. 

There  being  fewer  conditions  than  unknowns,  there  is 
an  unlimited  number  of  sets  of  values  of  the  unknowns 
which  might  satisfy  the  conditions,  and  we  have  to  de- 
termine from  the  n  observations  which  of  these  sets  is 
the  most  probable. 


ADJUSTMENT   OF   OBSERVATIONS  93 

Though  the  m  conditions  do  not  give  the  values  of  the  / 
unknowns,  they  enable  us  to  express  m  of  the  unknowns 
rigorously  in  terms  of  the  remaining  ones;  and  if  we  now 
substitute  these  expressions  for  the  m  unknowns  in  the 
observation  equations,  the  latter  may  then  be  adjusted 
for  the  most  probable  values  of  the  /  —  m  quantities  re- 
maining. The  most  probable  values  of  the  m  replaced 
quantities  may  now  also  be  calculated  so  that  the  condi- 
tions are  exactly  satisfied. 

Applying  this  to  the  case  of  the  angles  of  a  triangle, 
subject  to  one  condition  (47),  one  of  the  angles,  say  gs, 
may  be  expressed  by  means  of  it  in  terms  of  the  other 

^^''''  gs  =  180°  -  9i  -  ^2.  (48) 


The  three  observation  equations  then  appear: 

(49) 


^2  =   -^2, 

180°  -q,-q2  =  53. 


Let  the  student  adjust  these  and  show  that  the  most 
probable  values  sought  are 

gi  =  ^i  +  i[180°- (51  +  ^2  4-^3)], 

^2  =  ^2  +  i[180°  -  (^1  +  ^2  +  Sz)],  (50) 

qs=Sz-\-i[lS0''-{si-\-S2  +  s,)], 

the  third  result  following  from  the  other  two  through  sub- 
stitution in  (48) ;  which  shows  that  the  results  sought 
can  be  obtained  by  adding  to  each  measured  angle  one- 
third  the  discrepancy  between  the  sum  of  the  measured 


94      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

angles  and  180°,  so  as  to  make  the  sum  correct.  This  is 
on  the  assumption  that  all  three  of  the  measurements 
are  equally  trustworthy.  (See  Chap.  VII.)  The  same 
proceeding  is  to  be  followed  in  every  case  where  the 
observation  equations  represent  the  separately  measured 
values  of  the  unknowns,  while  the  one  equation  of  condi- 
tion rigorously  gives  their  sum.  Cases  like  this  are  of 
common  occurrence. 

If  the  sides  of  the  triangle  are  measured,  as  well  as  the 
angles,  there  will  be  six  observation  equations  (at  least), 
and  three  equations  of  condition.  Of  these  latter,  one 
will  be  the  same  as  (47),  the  other  two  arising  from  the 
requirements  of  trigonometry  as  to  sides  and  angles. 

A  case  of  special  importance  to  the  surveyor  is  the  ad- 
justment of  the  sides  and  angles  of  a  polygon  of  land.  In 
addition  to  whatever  measurements  are  made  upon  the 
lengths  and  bearings  of  the  sides,  there  are  two  rigorous 
conditions  to  be  fulfilled,  namely  :  that  the  algebraic  sum  of 
the  projections  of  the  sides  on  an  east-and-west  line  is  zero, 
and  the  algebraic  sum  of  their  projections  on  a  north-and- 
south  line  is  zero.  This  adjustment  will  be  found  explained 
in  detail  in  the  more  advanced  works  on  plane  surveying. 

EXERCISES 

41.  1.  Draw  a  large  triangle  on  paper  with  a  fine  pencil, 
and  measure  with  a  protractor  each  of  the  angles.  Form 
the  observation  equations  and  the  equation  of  condition, 
and  from  them  deduce  the  most  probable  values  of  the 
angles. 


ADJUSTMENT   OF   OBSERVATIONS  95 

2.  Lay  off  on  a  straight  line  four  points,  A,  B,  C,  D. 
Measure  AB,  BC,  CD,  AC,  BD,  AD,  From  these 
measurements  form  observation  equations  and  compute 
the  most  probable  values  of  AB,  BC,  CD.  These  seg- 
ments may  be  conveniently  lettered  x,  y,  z. 

3.  The  following  measurements  were  made  upon  a  rec- 
tangular metallic  tank  to  determine  its  dimensions : 

Length  (inside)  27.31  cm. 

Width  (inside)  16.08  cm. 

Depth  (inside)  9.67  cm. 

Capacity  by  standard  graduates,  4.3217  liters. 

Find  the  most  probable  dimensions. 

4.  Draw  five  lines  radiating  accurately  from  a  common 
point  0,  the  further  extremities  being  A,  B,  C,  D,  E. 
Measure  with  a  protractor,  by  the  differential  method, 
and  turning  the  protractor  at  each  measurement,  each  of 
the  angles  AOB,  AOC,  AOD,  AOE,  BOC,  BOD,  BOB, 
COD,  COE,  DOE.  Determine  the  most  probable  values 
of  the  angles  AOB,  BOC,  COD,  DOE. 

5.  The  following  are  the  results  of  an  analysis  of  a  cer- 
tain medicinal  compound : 

Salts  of  calcium  1.26  per  cent. 
Salts  of  sodium  2.53  per  cent. 
Salts  of  iron  0.23  per  cent. 
Salts  of  manganese  0.14  per  cent. 
Salts  of  quinine  0.07  per  cent. 


96       THEORY   OF   ERRORS   AND   LEAST   SQUARES 

Salts  of  strychnine  0.02  per  cent. 
Water  95.67  per  cent. 

Find  the  most  probable  values  of  the  several  percentages. 

6.   Six  points,  supposed  to  lie  on  the  arc  of  a  circle,  have 
the  following  measured  coordinates : 


X 

y 

X 

y 

3.15 
2.67 
1.80 

2.49 
3.72 
4.69 

1.07 
-0.20 
-  1.84 

5.33 

5.98 
6.25 

Find  the  most  probable  coordinates  of  the  center  and 
most  probable  radius. 

7.   A  steel  tape  was  measured  under  different  conditions 
of  stretch  and  temperature,  as  follows : 


Centigrade  Temp. 

Tension,  Lb. 

Obs.  Length,  Ft. 

0° 

0 

100.031 

20 

10 

.064 

25 

8 

.068 

18 

12 

.063 

21 

15 

.069 

15 

15 

.062 

Using  the  approximate  formula 

l  =  lo  +  at-\-  bf, 

in  which  t  =  temperature   and  /  =  tension,   adjust  for 
the  most  probable  values  of  h,  a,  h. 


ADJUSTMENT   OF   OBSERVATIONS  97 

8.  (Adapted  from  Wright's  Adjustment  of  Observations.) 
Let  D  be  the  difference  in  length  of  two  standard  meter 
bars  at  62°  F.  and  A  the  difference  in  their  coefficients 
of  expansion.  Then  the  difference  d  in  length  at  any 
temperature  t  is      ^  =  D -\- (t  -  Q2)  A. 


Observations  were 

made 

as  follows : 

t 

d 

24°.7 

0.00791  inch 

37.1 

811  inch 

61.7 

833  inch 

49.3 

820  inch 

66.8 

847  inch 

71.5 

849  inch 

Adjust  for  the  most  probable  values  of  D  and  A. 

9.   Van  der  Waal's  equation  for  pressure  and  volume 
of  a  gas  at  absolute  temperature  T  may  be  put  in  the  form 

v'^TR  —  va  -{-  pv'^b  -\-  tib  =  pv^. 

The  measurements  of  Amagat  on  air  at  moderate  pressures 
and  at  16°  C.  (289°.  1  absolute)  were  published  as  follows : 


p  IN  CM.  Mercury 

pv 

76 

1.0000 

2000 

0.9930 

2500 

.9919 

3000 

.9908 

3500 

.9899 

4000 

.9896 

98        THEORY   OF   ERRORS   AND   LEAST   SQUARES 

Form  the  observation  equations  by  the  method  of  Note  B, 
Appendix,  and  adjust  for  a,  b,  R. 

10.  The  electrical  conductivity  of  selenium  is  found  to 
vary  with  the  intensity  of  light  falling  on  it  according  to 
the  equation  ,_ 

The  following  data  were  furnished  by  Dr.  F.  C.  Brown. 

Intensity  / 

0 

3 
11 
17 
25 

Adjust  for  the  most  probable  values  of  a  and  b.  (Note. 
—  In  working  the  above  problem,  it  will  be  found 
necessary,  as  is  sometimes  the  case,  to  use  caution  in 
dropping  decimal  places,  as  the  normal  equations  hap- 
pen to  be  quite  "  sensitive  "  to  slight  changes  in  the 
coefRcients.) 

11.  The  E.M.F.  of  a  thermo-couple  for  a  given  tem- 
perature difference  t  between  junctions  may  be  represented 

by  the  equation 

e  =  at-\-  btK 

The  following  values  for  a  copper-tellurium  couple  with 
one  junction  at  0°,  in  which  e  is  in  volts,  were  furnished 
by  Mr.  W.  E.  Tisdale. 


CONDU 
TIVITY 

c- 
C 

Intensity  / 

Conduc- 
tivity C 

83 

33 

319 

188 

44 

348 

250 

50 

361 

285 

100 

446 

303 

ADJUSTMENT   OF   OBSERVATIONS  99 


t 
50 

e 

t 

0.000243 

t 
150 

e 

1 

0.000254 

82 

242 

162 

262 

92 

245 

180 

265 

100 

248 

186 

267 

113 

249 

190 

266 

117 

250 

195 

267 

128 

252 

200 

268 

140 

251 

Calculate  the  most  probable  values  of  the  constants  a 
and  b.     Plot  the  curve. 

12.  In  a  sine  intensity  magnetometer,  let  the  pole 
strength  of  the  bar  magnet  be  P,  the  distance  between  its 
poles  /,  and  the  distance  from  its  center  to  the  needle  pivot 
a.  8  is  the  needle  deflection  and  H  the  horizontal  inten- 
sity of  the  earth's  magnetism.  The  equation  connecting 
these  quantities  is 


The  following 

data  were 

obtained  at  a 

station  where 

H  =  0.1884  (c. 

g.s.). 

a  (cm). 

5 

a  (cm). 

d 

20 

24°  17' 

40 

1°  49' 

25 

12  46 

45 

1  35 

30 

6  48 

50 

1  27 

35 

3  26 

Find  the  most  probable  values  of    P  and  /.     Use  the 
method  of  Note  B,  Appendix. 


100  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

13.  The  specific  volume  of  a  certain  liquid  was  meas- 
ured at  different  temperatures  by  a  quick  secondary 
method  which  was  known  to  have  certain  small  persistent 
errors.  At  three  of  the  temperatures,  known  as  "  tie 
points,"  the  specific  volume  was  also  measured  by  a  more 
laborious  absolute  method,  free  from  the  said  sources  of 
error.     The  results  follow: 


Temp.,  C. 

Sp.  Vol., 

Sp. 

Vol., 

Secondary 

Absolute 

23°.0 

0.952750 

23.5 

2879 

24.0 

3003 

24.5 

3177 

0.953322 

25.0 

3339 

3488 

25.5 

3505 

26.0 

3678 

26.5 

3840 

4059 

27.0 

4012 

27.5 

4187 

28.0 

4366 

28.5 

4526 

29.0 

4701 

29.5 

4872 

30.0 

5075 

31.0 

5426 

In  order  to  correct  all  the  data  in  the  second  column,  use 
was  made  of  the  equation 

Y  =  AX  -\-  B, 

in  which  X  is  the  specific  volume  by  the  secondary  method 
and  Y  the  corresponding  corrected  value,  A  and  B  being 
assumed  constant.     By  using  the  values  of  X  and  Y  at 


ADJUSTMENT   OF   OBSERVATIONS        ,      Iftl; 

the  three  "  tie  points,"  find  the  most  probable  values  of 
A  and  B  and  correct  all  the  secondary  data  accordingly. 

14.  A  glass  sinker,  used  for  precision  measurements 
of  liquid  density  by  the  Archimedes  buoyancy  method, 
was  calibrated  for  expansion,  the  data  being  as  follows : 


Temp.  C. 

Vol.  Sinker  (cc.) 

24°.0 

34.03894 

24.5 

3907 

25.0 

3984 

25.5 

4085 

26.0 

4102 

26.5 

4134 

27.0 

4191 

27.5 

4203 

28.0 

4231 

28.5 

4240 

29.0 

4290 

30.0 

4393 

Find  the  most  probable  zero  volume  and  coefficient  of  ex- 
pansion of  the  sinker,  assuming  a  linear  relation. 

16.   Guthe  and  Worthing's  formula  for  the  vapor  pres- 
sure of  water  at  temperature  ^  C.  is 

logio  y  =  7.39992  - 


(^  +  273)' 


From  the  following  data,  find  the  most  probable  values  of 
a  and  h. 


10:^      THmRY   OF   ERRORS   AND    LEAST   SQUARES 


t 

p   (mm.  of  Mercury) 

10° 

9 

20 

17 

30 

32 

40 

55 

50 

92 

60 

149 

80 

355 

00 

760 

16.  The  angles  and  the  sides  of  a  triangle  ABC  were 
measured,  with  the  following  results. 

Angle  A    51°  9' 
5    95    4 
C    33  51 

Side  5(7    1721.3  ft. 
AC     2207.5 
AB     1233.0 

Introduce  the  necessary  geometric  conditions  and  adjust 
for  the  most  probable  values  of  sides  and  angles. 

17.  Draw  accurately  a  large  quadrilateral  ABCD  and 
its  two  diagonals  AC  and  BD.  Measure  with  a  millimeter 
scale  the  four  sides  AB,  BC,  CD,  DA,  and  with  a  protrac- 
tor the  angles  DAB,  DAC,  CAB,  ABC,  ABD,  DBC, 
BCD,  BCA,  ACD,  CDA,  CDB,  BDA.  Introduce  the 
eight  necessary  geometric  conditions  and  adjust  for  the 
most  probable  values  of  the  sides  and  angles  of  the  quad- 
rilateral. 


ADJUSTMENT   OF   OBSERVATIONS  103 

If  the  diagonals  had  also  been  measured,  how  many 
conditions  would  be  introduced  ? 

18.  Adjust  the  transit  observations  given  in  Art.  38. 

19.  (Adapted  from  Crandall's  Geodesy  and  Least 
Squares.)  Adjust  the  following  transit  observation  equa- 
tions for  X,  y,  z. 

-  0.07 X  +  1.41  y  -{-z  =  -  0.65, 
+  0.68  X  +  1.00  y  +  2  =  +  0.18, 
+  0.52  X  +  1.02  2/  +  2  =  +  0.13, 
+  2.51  X  -  2.67  2/  +  2  =  +  3.96, 

-  0.73  X  +  2.13  y  +  z  =  -  1.88, 
+  0.75  X  +  1.01  y-\-z=-\-  0.02, 
+  0.53  X  +  1.02  y  -\-z  =  -\-  0.13, 
+  0.68  X  +  1.00  y  +  z  =  +  0.44, 
+  0.81  X  +  1.02  2/  +  2  =  +  0.29, 
+  0.09  X  +  1.27  y  -\-z  =  -  0.76. 

20.  In  the  following  observation  equations  the  un- 
knowns n  and  K  are  constants  of  metallic  reflection. 


4'n-=iiTs^]-"^*- 


Assuming  that  approximate  values  of  n  and  K  are  known, 
transform  these  into  observation  equations  of  the  first 
degree  by  the  method  of  Note  B,  Appendix. 


CHAPTER  VI 
EMPIRICAL   FORMULAS 

42.  Classification  of  Formulas.  —  If  we  examine  into 
the  many  formulas  employed  to  represent  natural  or 
physical  laws,  it  is  found  that  they  fall  into  two  fairly 
distinct  classes,  which  may  be  called,  respectively,  rational 
and  empirical  formulas. 

To  the  former  class  belong  those  which  have  been  de- 
duced through  processes  of  mathematical  reasoning  from 
the  elementary  and  established  laws  of  the  science  to 
which  they  pertain;  hence  the  term  rational.  Such,  for 
example,  are  the  equations  for  the  motion  of  falling  bodies, 
the  expressions  for  electric  or  gravitational  force  at  a 
point,  the  equations  of  the  balance,  the  error  equations 
for  the  astronomical  transit,  etc.  In  these  there  appear 
certain  constants  or  coefficients,  the  determination  of  which 
is  often  a  matter  of  great  scientific  importance. 

Empirical  formulas,  on  the  other  hand,  are  those  whose 
form  is  inferred  wholly  from  the  results  of  experiment 
or  observation,  and  which  have  not  been  deduced  theoreti- 
cally. Some  of  the  best  examples  of  these  are  to  be  found 
in  engineering,  such  as  the  formulas  for  the  flow  of  water 
in  pipes  and  channels,  or  for  steam  pressure  as  a  function 
of  temperature.     Empirical  formulas  also  contain  con- 

104 


EMPIRICAL    FORMULAS  105 

stants,  which  are  determined  in  exactly  the  same  manner 
as  if  the  formulas  were  rational,  and  whose  determination 
depends  upon  experiment  and  measurement. 

A  closer  examination  into  the  subject  reveals,  however, 
the  fact  that  the  boundary  between  these  two  classes  is  by 
no  means  a  sharp  one,  for  the  reason  that  a  very  large 
proportion  of  the  rational  formulas  purporting  to  represent 
natural  laws  have  been  deduced  upon  more  or  less  em- 
pirical and  approximate  assumptions,  which  have  been 
adopted  for  the  sake  of  simplicity  of  form,  or  for  want  of 
better  information.  In  fact,  it  may  well  be  doubted 
whether  there  exist  any  absolutely  rational  formulas  per- 
taining to  material  magnitudes.  Even  Newton's  great 
law  of  gravitation  has  its  experimental  basis;  and  it  is 
possible  that  some  future  investigation  in  astronomy  may 
demonstrate  it  to  be  inaccurate. 

43.  Uses  and  Limitations  of  Empirical  Formulas.  — 
Empirical  formulas  owe  their  existence  to  the  fact  that  in 
many  cases  no  rational. formula  can  be  deduced  to  repre- 
sent the  law  of  behavior  of  a  phenomenon,  but  that, 
nevertheless,  experiment  shows  some  law  is  being  obeyed 
which  appears  to  be  simple  in  character  and  is  therefore 
presumably  expressible,  at  least  approximately,  in  mathe- 
matical symbols.  Not  being  able  to  trace  the  mechanism 
operating  between  cause  and  effect,  on  account  of  its  com- 
plexity or  for  other  reasons,  the  experimenter  must  seek 
more  or  less  blindly  for  a  functional  relation  that  will 
satisfactorily   connect   them.     It   may  happen  that  the 


106      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

finding  of  such  a  relation  as  accords  perfectly  with  the 
observations  will  throw  much  light  on  the  nature  of  the 
mechanism  itself,  and  lead  to  a  theory  relative  to  it,  which 
can  be  tested  by  more  intelligently  directed  later  ex- 
periments. Stefan's  fourth-power  law  of  cooling,  which, 
though  wholly  empirical  as  far  as  Stefan  was  concerned, 
has  led  to  the  important  modern  theory  of  radiation,  is 
an  excellent  example  of  this  sort. 

But  the  great  majority  of  empirical  formulas  are  con- 
fessedly artificial,  and  reveal  nothing  of  the  real  nature  of 
the  connection  between  the  phenomena  involved.  Many 
do  not  even  pretend  to  consistency  in  the  matter  of  dimen- 
sions ;  the  writer  has  estimated  railroad  culvert  openings, 
for  example,  on  the  crude  working  rule  that  the  area  of 
opening,  in  square  feet,  should  be  equal  to  the  square  root 
of  the  drainage  area,  in  acres  —  an  area  equal  to  a  length. 

Nevertheless  these  formulas  are  capable  of  the  utmost 
practical  usefulness;  for  by  means  of  them,  depending 
upon  the  principle  of  continuity,  we  may  accurately  inter- 
polate the  values  of  the  unknown  function  between  points 
actually  observed,  and  even,  in  a  limited  way,  extrapolate 
beyond  the  experimental  region  into  conditions  unattain- 
able in  practice. 

There  is  still  another  class  of  empirical  formulas,  more 
or  less  in  the  nature  of  scientific  curiosities,  which  repre- 
sent, in  the  experimental  region  only,  a  relation  between 
variables  that  have  no  conceivable  connection  with  each 
other.  It  is  thus  possible  to  construct  an  artificial  for- 
mula which  will  follow,  with  fair  accuracy,  the  increase 


EMPIRICAL    FORMULAS  107 

in  population  of  the  United  States,  or  of  a  city,  with  time, 
or  even  the  fluctuations  of  the  stock  market  over  a  given 
interval  of  time.  Such  formulas  are,  however,  of  little 
value,  as  they  are  merely  a  sort  of  cast  of  a  series  of  statis- 
tics which  are  themselves  available ;  and  since  the  variable 
represented  may  not  even  be  continuous,  interpolation 
and  extrapolation  with  any  certainty  are  impossible. 

It  must  also  be  pointed  out  that  empirical  formulas 
cannot  be  allowed  to  enter  into  theoretical  developments 
on  the  same  basis  as  rational  ones,  unless  their  physical 
nature  is  first  carefully  looked  into  and  the  region  in  which 
they  are  assumed  to  apply  is  properly  circumscribed. 
Where  the  true  functional  relation  (supposing  one  to 
exist)  can  be  dealt  with  mathematically  with  safety,  an 
artificial  one  closely  approximating  it  may  lead,  if  so  used, 
to  altogether  erroneous  conclusions. 

44.   Illustrations  of  Empirical  Formulas. 

1.  Reduction  of  Pendulum  to  Zero  Arc.  —  The  Kater's 
reversible  pendulum  is  familiar  to  nearly  every  physical 
laboratory  student  as  a  means  of  obtaining  the  accelera- 
tion of  a  faUing  body,  g,  or  the  value  of  "  gravity."  When 
so  adjusted  that  the  time  of  swing  is  the  same  from  both 
supports,  i.e.,  when  the  knife  edges  are  at  conjugate 
points,  the  pendulum  swings  in  a  period  given  by  the  ideal 
simple  pendulum  formula 


in  which  /  is  the  distance  between  the  knife  edges.     The 


108  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

determination  of  g  is  therefore  a  matter  of  measuring  the 
period  of  oscillation  and  the  distance  I. 

This  equation  affords,  however,  an  excellent  example  of 
the  class  referred  to  in  the  last  paragraph  of  Art.  42.  For 
in  its  deduction  it  is  assumed  that  the  pendulum  swings 
without  any  kind  of  friction  from  a  perfectly  rigid  support, 
and  that  the  amplitude  of  vibration  is  infinitely  small, 
none  of  which  conditions  is  attainable.  The  writer  has 
attacked  these  difficulties  in  the  following  manner,  with 
good  results. 

Apparatus  is  arranged  to  release  the  pendulum  so  as  to 
swing  with  any  desired  initial  angle  of  amplitude,  and 
the  time  accurately  observed  for  each  of  several  small 
amplitudes.  The  following  results,  obtained  by  one  of 
my  students,  are  typical.  </>  is  the  half  amplitude  in 
degrees,  T  the  period  in  seconds. 


«^ 

T 

<}> 

T 

1° 

0.878489 

6° 

0.878807 

2 

8543 

7 

8874 

3 

8622 

8 

8938 

4 

8679 

9 

8975 

5 

8740 

10 

9029 

The  steady  increase  of  period  with  amplitude  includes 
all  factors :  the  true,  theoretical  increase  that  would  exist 
under  ideal  conditions,  and  the  influences  of  air  friction, 
pivot  friction  and  bending  of  supports.  The  results 
are  plotted  in  Fig.  8,  which  shows  an  unmistakable  cur- 


EMPIRICAL   FORMULAS 


109 


vature  with  downward  concavity.  The  relation  is  there- 
fore not  Hnear,  but  may  be  approximately  quadratic. 
The  empirical  formula 

T  =  a  +  b<t> +  €<!>''  (51) 

is  now  assumed  to  represent  the  variable  T  as  a  function  of 
<f).     This  is  treated  as  a  form  of  observation  equation  in 


O 
0) 

y 

O.i 

< 

^ 

r 

I 

y 

y" 

Y" 

< 

y 

Y 

/ 

Y 

/> 

Y 

^ 

r\f 

Tao 

)          1          2         S        4         5         6         7         8         9        lO        <i,"» 

1           1           1           1           1           1          1           1           1           1           1 

i 

Fig.  8 

which  the  coefficients  a,  6,  c  are  the  unknowns.  The  ob- 
servation equations  and  residuals  are  written  out  as  usual, 
the  normal  equations  deduced  and  solved,  with  the  ap- 
proximate results 

a  =  +  0.878400, 

6  =  +  0.000073, 
c  =  -  0.000001, 


110      THEORY   OF   ERRORS    AND   LEAST   SQUARES 

which  give 

T  =  0.878400  +  0.000073  <t>  -  0.000001  <j>^       (52) 

as  the  relation  desired.  In  reahty  only  a  is  wanted,  for  it 
is  the  value  of  T  for  zero  amplitude  that  we  are  seeking ; 
that  is,  the  limit  approached  by  T  as  0  approaches  zero. 
This  is  0.878400  sec.  By  this  slight  extrapolation,  there- 
fore, it  is  possible  to  extend  the  experiments  into  a  region 
unattainable  otherwise.  The  value  of  g  may  now  be 
calculated  from  this  result  and  the  measured  distance 
between  knife  edges. 

2.  Solubility  Formula.  —  Previous  to  the  theoretical 
calculation  of  a  rational  formula  for  solubility  in  terms  of 
temperature  (Art.  39),  the  relation  was  represented  by  an 
empirical  formula  of  simple  power  terms : 

s  =  a  +  bt  -\-  cf  +  dtK  (53) 

The  data  given  in  Art.  39  will  suffice  for  the  determination 
of  the  constants  a,  6,  c,  d,  a  result  being  obtained  which 
will  fit  the  observations  nearly,  if  not  quite,  as  well  as 
the  rational  expression.  This  exercise  is  left  to  the  stu- 
dent. 

3.  Gordons  Formula  for  Rectangular  Columns.  —  The  ul- 
timate strength  of  a  rectangular  column  under  compres- 
sion is  found  to  depend  fundamentally  upon  how  slender 
it  is ;  specifically,  upon  the  ratio  of  its  length  to  its  shorter 
transverse  dimension.  For  long,  slender  columns,  the 
relation  is  found  to  be  expressed  satisfactorily  by  the 
following  formula,  in  which  U  is  the  ultimate  compressive 


EMPIRICAL   FORMULAS 


111 


strength  per  square  inch  of  cross  section,  and  R  the  ratio 
of  length  to  least  width. 


U  = 


(54) 


a  and  h  are  the  empirical  constants  to  be  determined.  The 
following  data  refer  to  white-oak  timber  columns  or  posts, 
U  being  expressed  in  pounds  per  square  inch,  and  will 
serve  as  a  further  exercise  for  the  student. 


R 

u 

R 

u 

10 

845 

25 

585 

12 

820 

28 

540 

15 

770 

30 

510 

18 

715 

35 

435 

20. 

675 

38 

400 

22 

640 

40 

375 

a  and  h  should  be  about  925  and  0.00091,  respectively. 

45.  Choice  of  Mathematical  Expression.  —  The  reader 
will  now  wish  to  know  by  what  process  the  form  to  be 
used,  as  representing  the  unknown  relation  between 
variables,  may  be  arrived  at.  There  is  no  general  rule 
covering  this  matter.  The  empirical  form  once  being 
settled  upon,  the  calculation  of  the  empirical  constants 
is  a  direct  process;  but  the  selection  of  a  mathematical 
expression  which  can  be  made,  by  the  use  of  proper  con- 
stants, to  fit  the  facts  with  sufficient  accuracy,  is  often  a 
problem  calling  for  the  exercise  of  the  highest  degree  of 
ingenuity,  especially  where  there  is  more  than  one  in- 
dependent variable. 


112  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

The  first  step  will  probably  always  be  to  plot  the  results 
of  the  observations,  or  the  data  to  be  represented,  to  a 
suitable  scale  on  coordinate  paper.  The  result  will  be 
some  sort  of  curve,  which,  if  at  all  regular,  will  give  an 
idea  as  to  the  nature  of  the  variation,  and  will  often  sug- 
gest an  equation  through  its  resemblance  to  some  well- 
known  locus,  such  as  the  straight  line,  parabola,  etc.  A 
few  general  forms  have  been  found  especially  adaptable. 

The  equation 

y  =  a  -\-  bx  -\-  cx^  -{-  dx^  +  •••,  (55) 

continued  so  far  as  may  be  necessary,  may  be  used  for 
curves  which  are  not  periodic,  nor  asymptotic,  nor  very 
irregular.  The  number  of  terms  to  be  used  will  be  limited 
by  the  fact  that  the  coefficients  of  powers  that  may  be 
omitted  turn  out  to  be  negligibly  small.  This  form  was 
used  in  two  of  the  three  examples  in  the  preceding  article, 
and  might  probably  have  been  used  with  some  success 
in  the  third.  It  was  remarked  in  connection  with  the 
second  example  of  Art.  36  that  the  volume  coefficient  of 
expansion  really  varies  when  carried  over  a  considerable 
range.  This  might  have  been  allowed  for  by  adding  a  term 
involving  the  square  of  t,  with  a  third  unknown  constant 
coefficient,  to  those  used.  When  such  a  form  as  (55)  is 
used,  it  will  be  well  to  apply  to  it  the  values  of  x  and  y 
belonging  to  five  or  six  of  the  observed  points  that  seem 
to  lie  most  accurately  on  the  curve,  as  a  preliminary 
calculation,  and  determine  from  them  approximate  values 
of  an  equal  number  of  the  coefficients  without  least  squares, 


EMPIRICAL    FORMULAS  113 

in  order  to  ascertain  where  the  series  may  safely  be  stopped. 
It  may  be  found,  in  this  way,  that  only  two  or  three  terms 
are  necessary,  the  coefficients  beyond  these  being  negli- 
gible. 

The  following  equations  are  also  quite  adaptable  to 
many  physical  phenomena,  particularly  those  involving 
variables  which  approach  a  limit,  or  in  which  maxima 
and  minima  do  not  appear. 

y  =  a  +  h\og{x  +  K).  (56) 

a^  =  a  +  6  log  (2/  +  K).  (57) 

ax  -\-  by  -\-  c  =  xy.  ,  (58) 

log  y  =  a  -\-  b  log  x.  (59) 

(56)  is  asymptotic  in  the  y  direction,  (57)  in  the  x  direc- 
tion; (58)  is  an  equilateral  hyperbola,  asymptotic  in 
both  directions.  The  logarithmic  formulas  may  easily 
be  put  into  exponential  form  if  desired.  The  constant  K 
may  sometimes  be  theoretically  assigned. 

The  equation 

ax  -\-  by  -\-  c  =  x'^y  (60) 

is  more  general  and  includes  (58).  (54)  will  also  be  seen 
to  be  a  special  case  of  it.  The  curves  represented  by  (60) 
may  have  maxima  or  minima  and  points  of  inflection. 
n  may  be  given  a  small  integral  value,  as  1,  2,  3,  and 
a,  b,  c  will  be  the  empirical  constants  to  be  determined. 
The  student  will  do  well  to  plot  these  equations,  using 
assumed  values  of  the  constants. 

For  functions  that  are  apparently  periodic,  or  which 


114     THEORY   OF   ERRORS   AND    LEAST   SQUARES 

have  many  "  ups  and  downs  "  in  the  course  of  the  varia- 
tion, there  may  be  used  a  Hmited  number  of  terms  of  the 
trigonometric  series 

y  =  a  +  b  sinnx  +  c  cos  nx 

-\-  d  sin  2  nx  -\-  c  cos  2  nx 

+  /  sin  3  nx  +  g  cos  3  nx  -\-  •■'  .     (61) 

This  is  a  Fourier's  series,  and  can  be  made  to  fit  any  curve 
with  any  desired  degree  of  approximation  by  carrying 
it  to  a  sufficient  number  of  terms.  The  calculation  of 
the  constants  may  become  extremely  laborious,  and  Prof. 
A.  A.  Michelson  devised,  some  years  ago,  a  mechanism 
known  as  the  harmonic  analyzer,  which  will  give  their 
approximate  values.  By  the  aid  of  this  machine  it  is 
possible  to  analyze  very  complicated  phenomena,  such  as 
the  tides  or  the  variations  in  terrestrial  magnetism,  into 
harmonic  components,  and  often  to  reveal  their  component 
causes.  But  it  is  also  possible,  by  this  means,  to  express 
in  an  altogether  artificial  manner  such  phenomena  as 
are  referred  to  toward  the  close  of  Art.  43.  To  empirical 
formulas  of  this  class  applies,  particularly,  the  caution 
against  treating  them  on  the  same  basis  as  rational 
formulas  in  mathematical  analysis. 

Very  often  the  problem  of  selecting  the  proper  form  will 
be  facilitated  by  giving  attention  to  obvious  limiting  con- 
ditions, such  as  the  fact  that  eftect  is  zero  where  cause  is 
zero,  etc.  This  amounts  to  making  the  selection  partly 
rational,  and  only  emphasizes  the  statement  that  there  is 
no  sharp  distinction  between  rational  and  empirical  ex- 


EMPIRICAL    FORMULAS 


115 


pressions.  After  all  has  been  said,  however,  the  student 
will  still  find  true  the  remark,  previously  made,  that  this 
matter  calls  for  skill  and  ingenuity  of  a  high  order. 


EXERCISES 

46.  1.  Experiments  were  made  upon  the  index  of  re- 
fraction of  a  solution  of  varying  concentration  and  density, 
sodium  light  being  used.     The  results  follow : 


Density  x 

Index  y 

Density  x 

Index  y 

1.200 

1.378 

1.146 

1.365 

1.187 

1.374 

1.132 

1.361 

1.178 

1.371 

1.123 

1.359 

1.167 

1.369 

1.115 

1.356 

1.156 

1.367 

1.098 

1.352 

Express  the  variation  by  a  suitable  empirical  formula, 
deducing  the  constants.  Would  it  be  safe  to  infer  from 
this  formula  the  index  for  pure  water? 

2.  A  galvanometer  attached  to  a  thermo-electric  couple 
gave  the  following  readings  y,  for  the  corresponding 
differences  of  temperature  x: 


X 

y 

X 

y 

0° 

0 

45° 

5.50 

20 

2.50 

50 

6.15 

25 

3.10 

60 

7.60 

30 

3.70 

70 

8.65 

35 

4.30 

80 

9.90 

40 

4.90 

hrf— 

90 

10.90 

Prepare  suitable  empirical  formula,  deducing  constants. 


116      THEORY   OF   ERRORS   AND   LEAST   SQUARES 
3.   The  following  are  observed  positions  of  points  on  a 


curve : 


X 

y 

X 

y 

0 

0 

5 

15.0 

1 

0.5 

6 

23.0 

2 

2.5 

7 

31.0 

3 

6.0 

8 

40.5 

4 

10.5 

9 

51.5 

Obtain  an  equation  whose  graph  will  fit  these  points  as 
nearly  as  possible,  and  plot  it. 

4.  The  temperature  of  a  heated  body,  cooling  in  the 
air,  was  taken  each  minute  for  ten  minutes,  the  results 
being  here  tabulated : 


Time  t 

.Temp.  0 

Time  t 

Temp.  0 

0 

84^9 

6 

6r.9 

1 

79.9 

7 

59.9 

2 

75.0 

8 

57.6 

3 

70.7 

9 

55.6 

4 

67.2 

10 

53.4 

5 

64.3 

The  temperature  of  the  air  was  20°.     Deduce  an  equation 
expressing  6  in  terms  of  t. 

6.  Measurements  were  made  upon  the  radioactivity 
of  a  deposit  of  pure  thorium  at  intervals  after  its  forma- 
tion, as  follows : 


EMPIRICAL    FORMULAS 


117 


Time 

Activity 

Time 

Activity 

10  min. 

100.0 

5hr. 

107.0 

20 

104.3 

6 

101.1 

40 

110.8 

8 

89.1 

60 

115.8 

10 

78.3 

80 

118.2 

12 

68.7 

100 

119.6 

15 

56.6 

120 

119.8 

18 

46.2 

3hr. 

117.9 

20 

40.7 

4 

113.0 

Express  the  relation  as  an  empirical  formula, 
activity  will  die  out  with  time.) 


(Note  that 


6.  The  quantity  of  discharge  Q,  in  cubic  feet  per 
minute,  of  a  10-inch  sewer  pipe  was  found  to  vary  with 
the  slope  (percentage  grade)  s  as  per  the  following 
data: 


s 

Q 

s 

0 

0.1  % 

64 

2.0% 

146 

0.2 

75 

3.0 

170 

0.4 

88 

4.0 

190 

0.6 

95 

5.0 

208 

0.8 

108 

10.0 

279 

1.0 

116 

Work  out  an  empirical  formula  and  plot  it. 

7.  The  means  of  many  observations  upon  a  certain 
variable  star  of  short  period  gave  the  following  variations 
of  magnitude ; 


118     THEORY   OF   ERRORS   AND   LEAST   SQUARES 


Time  (Days) 

Mag. 

Time  (Days) 

Maq. 

0 

4.65 

8    ' 

4.20 

1 

4.10 

9 

3.57 

2 

3.50 

10 

3.70 

3 

3.80 

11 

3.93 

4 

4.00 

12 

4.07 

5 

4.10 

13 

4.35 

6 

4.40 

14 

4.64 

7 

4.65 

15 

4.40 

Represent    this   variation   by   as   simple   a   formula   as 
possible. 

8.  The  atmospheric  refraction  R  for  a  star  above  the 
horizon  at  various  altitudes  a  is  given  approximately 
by  the  following  table,  corresponding  to  temperature 
50°  F.  and  normal  pressure : 


a 

R 

a 

R 

0'' 

34'  50" 

10° 

5'  16" 

2 

18  6 

20 

2  37 

4 

11  37 

40 

1  9 

6 

8  23 

60 

0  33 

8 

6  29 

90 

0  0 

Represent  these  as  nearly  as  possible  by  means   of   an 
empirical  formula. 


9.   Amagat's  experiments  on  air  at  very  high  pressures 
gave  the  following  results : 


EMPIRICAL    FORMULAS 


119 


Press.,  Atmos. 

Vol. 

Press.,  Atmos. 

Vol. 

1 

750 

1000 

•      1500 

1.000000 
0.002200 
0.001974 
0.001709 

2000 
2500 
3000 

0.001566 
0.001469      - 
0.001401 

Represent  these  by  an  empirical  formula. 

10.  The  current  through  the  field  coils  of  a  certain 
dynamo  was  varied  and  the  voltage  generated  by  the 
machine  simultaneously  measured,  as  follows: 


Field  Current, 

Armature 

Field  Current, 

Armature 

Amps. 

Volts 

Amps. 

Volts 

0.000 

0.0 

1.416 

21.0 

0.472 

8.5 

1.650 

23.2 

0.709 

12.1 

1.888 

25.5 

0.943 

15.4 

2.125 

27.3 

1.180 

18.3 

2.360 

29.0 

Represent  these  by  an  empirical  formula. 

11.    The  specific   gravity  of  dilute  sulphuric  acid  at 
different  concentrations  is  given  in  the  following  table : 


CONC. 

(Per  Cent.) 

Sp.  Gray. 

CONC. 

(Per  Cent.) 

Sp.  Gray. 

5 

1.033 

30 

1.218 

10 

1.068 

35 

1.257 

15 

1.101 

40 

1.300 

20 

1.139 

45 

1.345 

25 

1.178 

50 

1.389 

Represent  these  by  an  empirical  formula. 


120     THEORY   OF   ERROUS   AND   LEAST   SQUARES 

12.  A  pyknometer  being  tested  for  evaporation  was 
allowed  to  stand  in  a  desiccator  and  weighed  at  intervals, 
as  follows : 

Sept.  30    3  :  15  p.m.      44.4226  grams 
4  :  00  P.M.  .4223  grams 


Oct.  2 

11:00  a.m. 

.3855  grams 

3 :  30  P.M. 

.3821  grams 

Oct.  3 

8 :  00  A.M. 

.3695  grams 

4 :  00  P.M. 

.3622  grams 

Find  the  most  probable  weight 

at 

noon  October  1. 

13.   Simultaneous  observations  were  made  upon  two 
connected  variables  x  and  y  with  the  following  results : 


X 

y 

X 

y 

26.5  • 

0.002442 

31.5 

0.005315 

27.0 

2571 

32.0 

5607 

27.5 

2582 

32.5 

6039 

28.0 

2885 

33.0 

6407 

28.5 

3165 

33.5 

6947 

29.0 

3500 

34.0 

7238 

29.5 

3738 

34.5 

7703 

30.0 

4311 

35.0 

8092 

30.5 

4548 

35.5 

8438 

31.0 

4991 

36.0 

8870 

Represent  these  by  an  empirical  formula. 

14.   Following  are  vapor  pressures,  in  mm.  of  mercury,  of 
methyl  alcohol  at  various  temperatures : 


EMPIRICAL    FORMULAS 


121 


t 

P 

t 

P 

0'' 

30 

35° 

204 

5 

40 

40 

259 

10 

54 

45 

327 

15 

71 

50  . 

409 

20 

94 

55 

508 

25 

123 

60 

624 

30 

159 

65 

761 

Represent  these  by  an  empirical  formula. 
15.   Assuming  the  form 

log  n  =  logiV  —  log a  log^ 


15 


15 


in  which  n  is  per  cent,  and  s  is  grade,  deduce  N  and  a  from 
the  data  of  Ex.  10,  Art.  30.     Plot  the  curve. 

16.  The  following  average  heights  and  weights  for  men 
35  to  40  years  of  age  were  compiled  by  the  medical  director 
of  the  Connecticut  Mutual  Life  Insurance  Co. 


Height 

Weight 

Height 

Weight 

5  ft.  0  in. 

131 

5  ft.  8  in. 

157 

1 

131 

9 

162 

2 

133 

10 

167 

3 

136 

11 

173 

4 

140 

6   0 

179 

5 

143 

1 

185 

6 

147 

2 

192 

7 

152 

1       ' 

200 

Represent  these  by  an  empirical  formula. 


122     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

17.   The  Society  for  the  Promotion  of  Engineering  Edu- 
cation reports  its  growth  in  membership  as  follows : 


Year 

No.  Members 

Year 

No.  Members 

1894 

156 

1905 

400 

1895 

188 

1906 

415 

1896 

203 

1907 

503 

1897 

226 

1908 

675 

1898 

244 

1909 

747 

1899 

251 

1910 

938 

1900 

266 

1911 

1040 

1901 

261 

1912 

1166 

1902 

275 

1913 

1291 

1903 

326 

1914 

1358 

1904 

379 

Try  to  calculate  the  most  probable  membership  in  1915 
from  these  data. 

18.  Try  to  represent  the  data  plotted  in  Ex.  8,  Art.  30, 
by  means  of  an  empirical  formula. 

19.  The  following  measurements  give  the  average 
length  of  the  head  in  schoolboys  at  different  ages  (West, 
Science,  Vol.  21,  1893) : 


Age 

Length  (mm.) 

Age 

Length  (mm.) 

5 

176 

14 

187 

6 

177 

15 

188 

7 

179 

16 

191 

8 

180 

17 

189 

9 

181 

18 

192 

10 

182 

19 

192 

11 

183 

20 

195 

12 

183 

21 

192 

13 

184 

Represent  these  by  an  empirical  formula. 


EMPIRICAL   FORMULAS 


123 


20.  Records  of  the  magnetic  declination  (departure  of 
compass  from  the  true  north)  at  25°  N.  lat.,  110°  W.  long, 
over  a  series  of  years  are  as  follows  (U.  S.  Mag.  Tables  for 
1905) : 


1840 

9°  28'  E. 

1875 

10°  24'  E. 

1845 

38 

1880 

25 

1850 

49 

1885 

25 

1855 

10  00 

1890 

26 

1860 

09 

1895 

30 

1865 

16 

1900 

36 

1870 

21 

1905 

48 

Represent  these  by  an  empirical  formula. 


CHAPTER  VII 
WEIGHTED   OBSERVATIONS 

47.  Relative  Reliability  of  Observations.  Weights.  — 
We  have  hitherto  regarded  each  one  of  a  set  of  several 
observations  as  having  been  made  with  equal  mechanical 
refinement,  care  and  'skill,  and  the  results  as  meriting, 
therefore,  the  same  degree  of  confidence.  This  assumption 
is  often,  however,  far  from  the  truth.  The  position  of  a 
star,  for  example,  as  measured  with  an  engineer's  transit, 
13  less  reliable  than  it  would  bs  if  measured  with  a  large 
meridian  circle ;  and  the  results  of  a  series  of  difficult 
observations  made  by  a  tired  research  worker  in  a  cold, 
drafty  laboratory  are  not  worth  as  much  as  a  similar 
series  made  by  the  same  person  when  rested  and  under 
favorable  conditions.  Again,  the  mean  of  a  long  series  of 
careful  observations  upon  a  quantity  is  certainly  of  more 
value  than  the  result  of  a  single  measurement  upon  the 
same  quantity. 

It  is  therefore  evident  that,  in  practical  work,  it  is 
necessary  to  employ  some  means  whereby  differences  in 
reliability  may  be  taken  into  account.  This  can  be  done 
by  using  a  method  of  adjustment  in  which  the  more 
trustworthy  results  are  allowed  to  have  more  influence 
upon  the  final  most  probable  values  than  the  less  reliable 

124 


WEIGHTED   OBSERVATIONS  125 

ones,  thus  giving  each  result  a  degree  of  prominence  pro- 
portional to  its  reliability. 

To  accomplish  this,  it  is  the  practice  of  observers  to 
assign  to  different  observations,  numbers,  which  are  sup- 
posed to  represent  their  relative  degrees  of  reliability, 
and  which  are  called  weights.  Thus  an  observation  to 
which  the  weight  3  has  been  assigned  is  considered  to 
merit  only  half  as  much  attention  in  the  adjustment  as  one 
with  the  weight  6 ;  etc. 

In  order  to  have  some  basis  of  estimation,  we  may  regard 
an  observation  of  given  reliability  as  being  equivalent  to 
the  mean  of  a  certain  number  of  observations  considered 
as  having  standard  or  unit  weight,  and  this  number  is 
the  weight  of  the  observation  in  question.  The  assign- 
ment of  the  weight  10  to  an  observation  means  that  in 
the  opinion  of  the  observer  the  result  is  as  trustworthy  as 
the  average  of  ten  observations  of  unit  weight.  Any 
standard  of  trustworthiness  may  be  taken  as  a  unit,  but 
it  should  be  such  as  to  render  the  weights  of  all  the  ob- 
servations referred  to  it  simple,  whole  numbers.  It  is  to 
be  remembered  that  weights  are  purely  relative  quantities. 

The  assignment  of  weights  to  the  several  observations 
of  a  set  is  a  task  demanding  the  exercise  of  skill  and 
careful  judgment.  If  each  observation  is  actually  the 
mean  of  several  elementary  observations  and  all  are  of 
the  same  kind,  the  matter  is  comparatively  simple,  since 
there  is  in  this  case  a  numerical  basis  of  estimate.  Other- 
wise, and  especially  when  the  observations  are  of  different 
kinds,  the  assignment  is  not  so  easy.     The  problem  pre- 


126     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

sents  many  analogies  to  that  of  giving  numerical  grades 
to  pupils. 

Like  other  processes  of  the  sort,  the  weighting  of  ob- 
servations cannot  be  covered  by  any  set  of  definite  rules. 
It  may  be  suggested  that  the  observer  should  note  and 
record  in  detail  the  peculiar  circumstances,  if  any,  attend- 
ing each  observation  or  set  of  observations  which  is  to 
enter  into  the  final  adjustment,  and  allow  no  source  of 
unusual  disturbance  to  go  unnoticed.  Often  it  is  well  to 
assign  weights  at  the  time  of  the  observation,  while  all 
the  circumstances  are  fresh  in  the  mind,  but  this  should 
not  take  the  place  of  recording  the  circumstances.  It 
sometimes  happens  that  some  one  else  examines  the  original 
notes  and  prefers  to  assign  weights  for  himself.  I  recall 
a  case  of  this  sort,  in  which  the  weighting  depended  solely 
upon  the  records  which  the  observer  had  kept  of  the 
weather  conditions  prevailing  at  the  time  of  each  experi- 
ment. This  was  because  wind  and  fluctuations  of  tem- 
perature were  causes  of  marked  disturbance  in  this  par- 
ticular work. 

48.  Adjustment  of  Observations  of  Unequal  Weight.  — 
In  adjusting  a  set  of  observations  to  which  different  weights 
have  been  assigned,  we  have  but  to  remember  that  the 
weight  w  signifies  that  the  observation  in  question  is  the 
equivalent  in  importance  of  w  observations  of  unit  weight. 
It  is  therefore  necessary  only  to  repeat  the  corresponding 
observation  equation  w  times,  and  then  proceed  as  usual 
with  the  reduction  to  normal  equations.     That  is,  if  the 


WEIGHTED   OBSERVATIONS  127 

first  observation  has  weight  2,  the  second  5,  the  third  3, 
etc.,  then  simply  write  the  first  observation  equation  twice, 
the  second  five  times,  the  third  three  times,  etc.  The 
number  of  observation  equations  is  now  Xw,  the  sum  of 
the  weights. 

A  simple  illustration  of  this  is  the  case  of  n  direct  ob- 
servations on  a  single  quantity  q.  If  the  results  are  ^i,  ^2, 
•••,  s„  with  weights  wi,  W2,  •••,  Wn,  the  most  probable  value 
as  deduced  on  the  above  principle  is 


WiSi  +  W2S2  + h  WnSn 

i^w  = T T T ' 

or  m^=  -^ — ^  (62) 


This  is  called  the  weighted  mean.     If  all  the  weights  are 
equal,  it  becomes  simply  the  mean. 

With  observation  equations  of  the  first  degree  involving 
several  unknowns,  the  process  can  be  effected  by  first 
multiplying  the  expression  for  each  residual  by  the  coeffi- 
cient of  the  unknown  contained  therein  (as  in  the  rule  at 
the  close  of  Art.  34),  then  multiplying  by  the  weight  of 
the  corresponding  observation,  adding  the  results  and 
equating  the  sum  to  zero,  to  form  the  normal  equation. 
In  this  way  each  residual  is  represented  in  each  normal 
equation  a  number  of  times  equal  to  its  weight.  The 
same  thing  may  be  attained  by  first  multiplying  each 
of  the  original  observation  equations  by  the  square  root 
of  its  weight  and  then  proceeding  with  the  reduction 


128     THEORY   OF   ERRORS   AND    LEAST   SQUARES 

as  usual.  These  square  roots  need  only  be  indicated,  by 
means  of  radical  signs,  as  they  will  disappear  on  re- 
duction. (Let  the  student  show  why  the  square  roots  of 
the  weights  should  be  thus  used,  and  not  the  weights 
themselves.) 

In  the  reduction  of  observations  upon  quantities  limited 
by  conditions  (Art.  40),  it  is  evident  that  the  equations  of 
condition  are  not  to  be  weighted,  but  only  the  observation 
equations.  In  the  process  of  adjustment,  the  weighting 
should  be  introduced  after  the  conditions  have  been 
involved  in  the  observation  equations,  but  before  the  re- 
duction of  the  latter  to  normal  equations.  Some  of  the 
following  examples  will  illustrate  this. 

EXERCISES 

49.  1.  Measurements  were  made  upon  the  segments 
of  a  line  AB,  formed  by  points  C,  D  upon  it,  as  follows : 

Mean  of  2  observations  on  AC  =  45.10  ft. 

Mean  of  3  observations  on  AD  =  77.96  ft. 

Mean  of  2  observations  on  CD  =  32.95  ft. 

Mean  of  3  observations  on  CB  =  98.36  ft. 

Mean  of  2  observations  on  DB  =  65.55  ft. 

Mean  of  4  observations  on  AB  =  143.55  ft. 

Find  the  most  probable  values  of  AC,  CD,  DB. 

2.  In  one  time-observation  with  a  transit  instrument, 
only  five  of  the  nineteen  lines  of  the  reticle  were  used,  viz., 
Nos.  2,  5,  10,  15,  .18.     A  second  observation  employed 


WEIGHTED   OBSERVATIONS 


129 


Pig.  9 


all  the  lines.  What  can  be  said  as  to  the  relative  weights 
of  the  two  observations,  the  method  of  observing  being 
the  same  in  both  cases?     (Fig.  9.) 

3.  In  determining  the  constants  of  a  balance,  it  was 
borne  in  mind  that  the  instrument  was  to  be  used  re- 
peatedly for  the  weighing  of  an  ob- 
ject varying  slightly  in  weight  but 
always  in  the  neighborhood  of  43  to 
45  grams.  Hence  the  sensibility  was 
measured  twenty-five  times  with  a 
load  of  45  grams,  giving  a  mean 
of  2.402  scale  divisions  per  milli- 
gram, and  only  four  times  with  zero  load,  giving  a 
mean  of  2.767  scale  divisions.  Determine  the  most 
probable  values  of  the  balance  constants  (Art.  35, 
Ex.2).  . 

4.  Draw  a  triangle  and  measure  its  angles  with  a  pro- 
tractor, one  angle  being  measured  but  once,  the  second 
three  times,  the  third  eight  times  (or  some  other  set  of 
unequal  numbers),  all  the  measurements  being  made 
differentially.  Introduce  the  necessary  condition,  assign 
the  proper  weights  and  deduce  the  most  probable  values  of 
the  angles. 

6.  The  following  pointings  were  made  at  three  sta- 
tions in  the  triangulation  of  California,  using  a  50- 
cm.  direction  theodolite  (U.  S.  Coast  Survey  Report, 
1904) : 


130     THEORY   OF   ERRORS   AND   LEAST   SQUARES 


Station 

Pointing  on 

Circle  Reading 

Wt. 

San  Pedro 

1  Wilson  Peak 
1  San  Juan 

73°  11'  40".97 

6.1 

118   57    57  .51 

6.1 

San  Juan 

1  San  Pedro 

16   54   50  .29 

7.6 

I  Wilson  Peak 

84   26   21  .03 

7.6 

Wilson  Peak 

1  San  Juan 

241    39   01  .29 

6.7 

1  San  Pedro 

308   21    21  .51 

6.7 

Adjust  for  the  most  probable  angles. 

6.  The  range  of  magnitude  of  the  variable  spectroscopic 
binary  star  a  Geminorum  was  measured  by  a  selenium 
photometer  on  different  nights  as  follows  (Stebbins, 
Astrophysical  Journal) : 


Range 

Wt. 

Range 

Wt. 

Range 

Wt. 

0.237 

5 

0.235 

3 

0.218 

4 

.217 

4 

.197 

5 

.233 

4 

.233 

5 

.217 

5 

.209 

3 

.231 

5 

.210 

5 

.224 

5 

.217 

5 

.222 

5 

.227 

5 

.205 

5 

.213 

5 

.189 

3 

.207 

5 

.223 

5 

.220 

5 

.227 

5 

.250 

5 

.211 

4 

.231 

5 

.219 

5 

Find  the  weighted  mean  of  these  observations. 


7.  Following  are  results  from  precise  leveling  in  Texas 
(U.  S.  Coast  Survey  Report,  1911).  The  weights  assigned 
are  inversely  proportional  to  the  squares  of  the  distances 
between  the  stations. 


WEIGHTED   OBSERVATIONS 


131 


Lavernia  above  Serita     . 
Thomas  above  Serita 
Serita  above  Stockdale   . 
Serita  above  Ruckman   . 
Stockdale  above  Ruckman 
Stockdale  above  Karnes 
Ruckman  above  Karnes 
Ruckman  above  Bryde  . 
Karnes  above  Bryde  .     . 
Ruckman  above  Choate. 
Bryde  above  Choate  . 
Bryde  above  Pettus    .     . 
Choate  above  Pettus  .     . 
Bryde  above  Barroum     . 
Pettus  above  Barroum    . 
Pettus  above  Wiess    .     . 
Choate  above  Wiess  .     . 


Meters 


Weight 


+  57.47 

4.8 

+  45.73 

1.0 

+  10.56 

2.9 

+  33.14 

0.6 

+  23.62 

1.1 

+  30.83 

0.6 

+    6.42 

1.8 

-11.83 

0.9 

-  18.66 

4.8 

+  20.17 

0.9 

+  32.65 

3.6 

+  23.34 

4.8 

-    9.36 

10.9 

+  11.51 

5.1 

-  11.71 

7.9 

+  26.19 

7.5 

+  17.40 

3.3 

Adjust  for  the  most  probable  elevations  above  the  lowest 
station  in  the  list. 

8.  Experiments  were  made  for  the  purpose  of  rating  a 
Price  current  meter,  used  in  measuring  the  velocity  of 
streams.  The  data  are  the  velocity  V  of  the  current  in 
feet  per  second  and  the  number  R  of  revolutions  per  second 
of  the  meter  (Raymond,  Plane  Surveying). 


V 

R 

Wt. 

V 

R 

Wt. 

3.774 

1.886 

2 

1.036 

0.466 

4.544 

2.295 

1 

1.105 

0.503 

4.878 

2.464 

1 

7.142 

3.678 

1.613 

0.774 

1 

2.740 

1.342 

1.316 

0.618 

1 

6.896 

3.552 

Assume  a  linear  relation  and  deduce  the  two  constants. 


132  THEORY   OF   ERRORS   AND   LEAST    SQUARES 

9.   Four  points,  y4, 5,  C,  D,  lie  consecutively  in  a  straight 
line.    The  following  distances  are  measured  with  a  steel  tape. 


AD     . 

.     2871.2  (Ave.  of  2) 

AB     . 

.     1042.0 

AC     . 

.     2433.5 

BC 1392.2 

BD 1828.6 

CD  . 437.5 


Apply  the  principle  that,  in  chaining,  the  weights  of 
similarly  measured  lines  are  inversely  proportional  to  the 
squares  of  their  measured  lengths,  and  adjust  the  above 
values  accordingly. 

10.  Zenith  telescope  observations  were  made  at  Roslyn 
Station,  Virginia,  upon  the  latitude  of  that  station  with 
various  pairs  of  stars,  as  follows  (Chauvenet,  Practical 
Astronomy).  The  weights  were  assigned  from  the  number 
of  observations  involved  and  the  precision  with  which  the 
declinations  of  the  stars  employed  had  been  measured. 


Observed  Lat. 

Wt. 

Observed  Lat. 

Wt. 

37"  14'  24"  .78 

0.44 

37"    14'  25".  15 

0.59 

25   .05 

.67 

25    .22 

.67 

24   .83 

.82 

24   .84 

.67 

26   .20 

.59 

25    .36 

.67 

25   .91 

.43 

26   .02 

.62 

22   .73 

.00 

25   .42 

.44 

25   .93 

.70 

26    .08 

.44 

25   .18 

.65 

25    .72 

.67 

25   .89 

1.09 

25   .70 

1.33 

25    .79 

1.33 

25   .93 

1.20 

24   .53 

0.29 

Find  the  most  probable  latitude. 


WEIGHTED   OBSERVATIONS 


133 


11.  (Adapted  from  Chauvenet,  Practical  Astronomy.) 
At  a  station  0  of  the  U.  S.  Coast  Survey,  angles  were  read 
on  each  of  four  other  stations,  A,  B,  C,  D,  as  follows: 


Angle 

Wt. 

Angle 

Wt. 

AOB        65°     ir    52 ".5 
BOC        66      24     15  .6 

3 
3 

COD          87°    2'   24"  .7 
DOA         141    21    21    .8 

3 

1 

Adjust  for  the  most  probable  angles. 

12.  Spectrographic  radial  velocity  measurements  were 
made  upon  the  Orion  nebula,  using  different  spectrum 
lines  on  different  dates,  as  follows  (Lick  Observatory 
Bulletin  No.  19) : 


Date 

Line 

Velocity  (Km.) 

Wt. 

Dec.    8,  1901 

Hy 

+  17.1 

3 

16 

Hp 

16.1 

2 

17 

Hp 

17.0 

2 

18 

Hy 

14.8 

3 

Find  the  most  probable  radial  velocity. 

13.   A   certain   critical   coefficient    of    expansion    was 
measured  several  times  with  different  apparatus. 


Observed  Val. 

Wt. 

Observed  Val. 

Wt. 

0.0045 
39 
34 
30 

3 

2 
5 
4 

0.0036 
26 

27 
43 

2 

2 
1 
3 

Find  the  most  probable  value  from  these  data. 


r34     THEORY   OF   ERRORS    AND   LEAST   SQUARES 

'  14.  The  following  data  are  right  ascension  corrections 
to  the  Berlin  Jahrbuch  made  by  the  photographic  transit 
at  Georgetown  Observatory  for  the  star  f  Ophiuchi  on 
different  dates. 


Cor. 

Wt. 

Cor. 

Wt. 

Cor. 

Wt. 

Cor. 

Wt. 

-  0.03  S. 

2 

+  0.02  S. 

2 

-0.01 

3 

+  0.02 

3 

-     .03 

3 

.00 

2 

-     .04 

2 

+     .02 

3 

-     .01 

1 

4-    .04 

1 

+    .03 

2 

.00 

2 

-    .02 

0 

-    .04 

1 

-    .02 

3 

-    .04 

3 

-    .03 

1 

-    .05 

1 

-    .06 

2 

-    .06 

3 

Find  the  weighted  mean. 

16.  (Adapted  from  Wright's  Adjustment  of  Observa- 
tions.) The  following  trigonometric  levelings  were  made 
between  two  terminal  stations  A  and  B,  as  follows : 


Stations 

Meters 

Wt. 

Stations 

Meters 

Wt. 

A    above 

12 

914.96 

23 

3  above  9 

216.46 

1 

A    above 

10 

1287.75 

17 

5  above  9 

899.87 

1 

A    above 

11 

1299.27 

2 

5  above  8 

1075.77 

1 

A    above 

9 

1553.09 

5 

3  above  8 

391.74 

1 

12  above 

10 

372.73 

5 

7  above  8 

901.78 

1 

12  above 

11 

384.41 

2 

5  above  7 

174.45 

7 

12  above 

9 

638.30 

3 

4  above  3 

296.69 

60 

12  above 

8 

814.35 

1 

7  above  3 

509.49 

4 

10  above 

11 

11.60 

3 

B  above  3 

1376.19 

14 

10  above 

9 

265.48 

6 

5  above  4 

387.24 

20 

10  above 

8 

441.10 

2 

7  above  4 

212.75 

7 

11  above 

9 

253.87 

1 

B  above  4 

1079.50 

30 

11  above 

8 

429.55 

10 

B  above  5 

692.35 

15 

9  above 

8 

175.37 

1 

WEIGHTED   OBSERVATIONS  135 

By  precise  spirit  leveling,  A  was  found  to  be  39.05  meters 
above  B,  which  may  be  taken  as  correct.  Adjust  the 
heights  of  the  other  stations  above  B  accordingly. 

50.  Wild  or  Doubtful  Observations.  —  It  sometimes 
happens  that,  in  the  course  of  a  series  of  measurements, 
results  occur  which  are  so  doubtful  that  the  observer  is 
tempted  to  reject  them  altogether.  In  technical  language, 
their  weight  is  so  small  as  to  be  seemingly  negligible,  and  it 
is  a  question  whether  their  retention  may  not  do  more 
harm  than  good. 

The  doubt  may  arise  from  the  existence  of  unusual  or 
disturbing  conditions,  known  to  the  observer.  On  one 
occasion  I  was  making  a  quantitative  analysis  to  determine 
the  exact  concentration  of  a  solution,  and  during  the  proc- 
ess of  drying,  accidentally  spilled  a  few  drops  of  hydrant 
water  into  the  residue.  My  final  result  was  to  be  an 
average  from  the  analyses  of  several  specimens,  and  the 
accident  would  unquestionably  vitiate  the  result  of  this 
observation ;  but  the  specimens  were  obtained  with  diffi- 
culty and  I  could  ill  afford  to  spare  any  of  the  data.  Was 
the  result  to  be  rejected  or  not  ? 

Again,  suspicion  may  be  due  to  a  marked  difference 
between  the  result  in  question  and  all  the  others  of  the 
set.  This  does  not  refer  to  mistakes  (Art.  9),  which  may 
usually  be  easily  rectified.  To  the  observer's  best  knowl- 
edge, the  doubtful  observation  deserves  as  much  weight 
as  the  others,  having  been  made  with  the  same  care ; 
but  he  dislikes  to  retain  it,  as  it  is  so  far  out  of  agreement. 


136     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

The  former  class  of  doubtful  observations  should,  in 
the  opinion  of  the  writer,  be  rejected  unless  some  idea  of 
the  extent  of  the  disturbance  can  be  obtained  and  due 
correction  made  for  it  if  necessary.  What  I  did  in  the  case 
cited  was  to  test  the  hydrant  water  and  ascertain  that 
the  amount  of  solids  contained  in  a  few  drops  would  not 
be  sufficient  to  affect  the  result  at  all  seriously;  but  I 
gave  only  half  as  much  weight  to  this  observation  as  to 
the  others. 

With  the  latter  class  the  case  is  more  doubtful.  Just 
because  a  result  differs  from  the  others  is  no  proof  that 
it  is  any  farther  from  the  truth,  especially  when  the  num- 
ber of  observations  is  small.  In  casting  out  such  a  result, 
one  may  be  throwing  away  his  most  valuable  observation. 
Certain  criteria  have  been  proposed  for  deciding  whether 
to  retain  or  reject  a  "  wild  "  observation,  based  upon  the 
law  of  error  distribution.  Probably  the  best  decision 
will  be  based  upon  the  observer's  judgment,  it  being  borne 
in  mind  that  results  of  observations  should  not  be  tampered 
with  unthinkingly.  Where  wide  deviations  occur,  it  will 
be  well,  if  possible,  to  continue  the  observations  until  a 
sufficient  number  are  accumulated  to  show  the  law  of 
distribution  with  some  distinctness  and  symmetry. 

51.  The  Precision  Index  h.  —  It  was  pointed  out  in 
Art.  28  that  the  quantity  h  in  the  error  equation  has  to 
do  with  the  precision  of  the  observations  (Art.  13),  and  that 
the  greater  the  value  of  h,  the  greater  is  the  precision  indi- 
cated,    h  may  thus  be  termed  the   ''  precision  index " 


WEIGHTED   OBSERVATIONS  137 

or  "  measure  of  precision."  We  are  here  naturally  led 
to  inquire  what  connection  exists  between  the  precision 
index  and  the  weight  of  an  observation.  For,  if  we  have 
two  sets  of  measurements,  one  of  which  is  more  precise 
than  the  other,  the  value  of  h  belonging  to  the  error  dis- 
tribution in  one  set  will  be  larger  than  that  belonging  to 
the  other ;  while  at  the  same  time  the  weight  of  one  ob- 
servation from  the  first  set  is  greater  than  that  of  one 
from  the  second  set. 

Let  hi  and  Ci  be  the  constants  in  the  equation  of  error 
distribution  corresponding  to  the  first  set,  and  let  Wi  be 
the  weight  of  an  observation  from  that  set,  supposing 
them  all  to  have  equal  weight;  and  let  fh,  C2,  W2  be  the 
corresponding  quantities  relating  to  the  second  set.  The 
probability  of  an  error  x  occurring  in  the  first  set  is 

2/1  =  cie-'''\  (63) 

Let  the  value  of  the  precision  index  corresponding  to  a  set 
in  which  the  observations  are  of  unit  weight  be  h.  This 
may  be  called  a  "  standard  index,"  though  no  absolute 
value  can  as  yet  be  assigned  to  it.  An  observation  from 
the  first  set  is  equivalent  in  worth  to  Wi  observations 
from  the  standard  set,  in  each  of  which  the  probability  of 


an  error  x  is 


-h^sii 


y  =  ce 

Therefore  the  probability  yi  of  the  error  x  occurring  in  the 
first  set  is  that  of  its  occurring  Wi  times-in  the  standard  set, 
which  is  y^\  giving 

y^  =  ym^^m^-w,h^x^^  (64) 


138     THEORY   OF   ERRORS   AND   LEAST   SQUARES 


The  error  x  being  supposed  the  same  in  (63)  and  (64),  and 
these  equations  holding  for  all  values  of  x,  comparison 
gives  at  once 


Likewise 


hi^  =  wih^. 
h<i^  =  Wih^, 


referring  to  the  observations  of  the  second  set,  having 
weight  W2.     That  is, 


hi^\  hi^'.hz'^:  '•'  =Wi:  i(;2 :  wz : 


(65) 


or  the  weights  of  ohservations  are  in  proportion  to  the  squares 
of  their  precision  indices. 

In  order  to  illustrate  this  principle,  let  the  error  distri- 
bution of  the  first  set  be  represented  by  A,  Fig.  10,  and 


Fig.  10 

that  of  the  second  by  5.  A  represents  the  more  precise  set 
of  measurements.  Let  the  points  of  inflection  be  distant 
a  and  b  from  the  ?/-axis  in  the  two  curves  respectively. 
From  (27),  Art.  28, 


a:b  = 


1 


/?iV2      h2^2 

or 

hi :  h2  =  b:  a. 

(66) 

Then  from  (65), 

wi :  W2  =  b^ :  a^. 

(67) 

WEIGHTED   OBSERVATIONS  139 

It  is  thus  possible,  by  means  of  a  study  of  residual 
curves,  to  estimate  the  relative  weights  of  observations 
made  by  different  processes,  or  with  different  instruments, 
or  by  different  observers.  In  the  next  chapter  (Art.  59) 
will  be  presented  a  mathematical  means  of  obtaining  the 
same  information,  without  plotting  the  curves.  This  does 
not,  of  course,  refer  to  the  weighting  of  individual  observa- 
tions of  the  same  set,  which  must  depend  upon  the  judg- 
ment of  the  observer  as  to  the  conditions  existing  at  the 
time. 

52.  General  Statement  of  the  Principle  of  Least 
Squares.  —  The  principle  of  least  squares,  already  enun- 
ciated in  three  ways  adapted  to  increasingly  complicated 
cases  of  adjustment  (Arts.  29,  31,  33),  may  now  be  deduced 
in  its  general  form,  which  includes  all  the  others  as  special 
cases. 

Let  a  series  of  ii  observations  be  made,  whose  weights 
are  respectively  Wi,  wi,  •••,  Wm  and  let  the  residuals  be 
Pu  Pi,  '",  Pn-     The  probabilities  of  these  residuals  are 

in  which  c  and  h  are  the  precision  constants  corresponding 
to  an  observation  of  unit  weight.  The  probability  of  the 
occurrence  of  all  of  this  particular  set  of  residuals  is 


140  THEORY   OF    ERRORS   AND    LEAST   SQUARES 

The  most  probable  set  of  residuals,  and  hence  those 
determined  by  the  most  probable  values  of  the  unknown 
quantities  involved  in  the  observations,  are  those  for  which 
y  is  a  maximum,  and  hence  those  for  which  11  (wp^)  is  a 
minimum. 

The  general  statement  follows :  The  most  probable 
values  of  unknoivn  quantities  connected  by  observation  equa- 
tions to  ivhich  weights  have  been  assigned  are  those  which 
will  render  the  sum  of  the  weighted  squares  of  the  residuals 
a  minimum.  The  meaning  of  the  term  ''weighted 
squares  "  is  obvious  from  the  above. 

The  rules  of  Art.  48  for  the  adjustment  of  weighted  ob- 
servations might  have  been  deduced  from  the  principle 
as  above  stated,  in  the  same  manner  as  the  deduction 
was  made  for  the  simpler  case  of  equal  precision  (Arts. 
33,  34). 


CHAPTER  VIII 

PRECISION   AND   THE   PROBABLE   ERROR 

53.  Discontinuity  pf  the  Error  Variable.  —  There  is 
one  point  in  the  foregoing  discussions  of  the  law  of  error 
that  has  not  been  emphasized.  In  all  of  the  mathematical 
work,  we  have  treated  the  error  as  if  it  were  a  true  con- 
tinuous variable  x,  which  might  have  any  value  whatever 
from  —  00  to  +  00 .  But  to  assume  this  would  be  to  assume 
an  infinitely  minute  graduation  of  our  measuring  scale. 
To  illustrate  the  fact,  let  us  suppose  that  the  measured 
quantity  is  an  angle.  If  the  error  were  a  continuous 
variable,  successive  measured  values  of  the  angle  need 
not  differ  by  so  much  as  a  billionth  of  a  second,  yet  might 
be  different ;  and  the  probability  of  any  particular  error 
out  of  the  infinity  of  possible  ones  would  be  infinitesimally 
small.  It  is  thus  seen  that  the  variable  error  x,  instead  of 
varying  by  infinitesimal  increments  dxy  really  has  equal 
finite  discontinuities  A,  which  represent  the  smallest 
fraction  of  a  unit  in  which  the  measured  results  are  ex- 
pressed. On  a  surveyor's  transit,  for  example,  A  is  usually 
one  minute  for  single  angle-readings;  w^hile  with  the 
micrometers  used  on  large  equatorial  telescopes,  angular 
measurements  are  made  which  may  be  expressed  in  hun- 
dredths of  a  second. 

141 


142      THEORY   OF   ERRORS   AND   LEAST   SQUARES 


/T 


/" 


\ 


The  error  curve  may  then  be  represented  as  a  sort  of 
stairway  with  equal  treads  and  unequal  risers,  and  the 
errors  considered  as  falling  into  compartments  correspond- 
ing to  the  several 
A's,  just  as  do  the 
shots  in  the  target 
experiment  of  Art. 
11. 

The  width  of  one 
of  these  error  "  com- 
partments "  being  A 
and  its  height  y,  it 
may  be  looked  upon  as  a  narrow  strip  of  finite  area  yA. 
y  is  the  probability  that  an  error  will  fall  into  the  com- 
partment to  which  the  ordinate  y  corresponds.  Let  us 
imagine  all  the  strips  placed  end  to  end,  the  total  length 
being  2i/  =  1,  since  this  is  the  sum  of  the  probabilities 
of  all  possible  errors  (Art.  16).  But  the  area  of  this  long 
strip  is  the  area  of  the  curve : 


Fig.  11 


a:=4-oo 


,-/l2x2 


and 


=  20^6-^'^'  '^) 


x  =  0 


^      A      A  ^ 

x  =  0 


(69) 


A  being  small,  the  summation  S  in  (69)  is  for  practical 
purposes  represented  by  the  definite  integral 


PRECISION  143 

/  =  Ce-^'^^'dx.  (70) 

Whence,  from  (69),  . 

c=A.  (71) 

The  integral  7  is  a  function  of  h,  and  when  we  have  de- 
termined what  function  of  h,  we  shall  have  found  the  re- 
lation between  the  two  constants  c  and  h  of  the  error  equa- 
tion, which  was  referred  to  in  Art.  28. 

54.  Value  of  the  Integral  /,  and  the  Relation  between  c 
and  h.  —  The  evaluation  of  the  definite  integral  I  of  the 
preceding  article  is  worked  out  in  Note  C  of  the  Appendix, 
this  being  a  problem  belonging  to  the  theory  of  definite 
integrals  and  an  interesting  example  of  a  method  often 
employed  in  such  cases.     The  result  is 

7=X^--<ix  =  ^.  (72) 

The  student  who  does  not  care  to  follow  out  the  proof  may 
verify  the  result  by  plotting  the  function  for  two  or  three 
chosen  values  of  h  and  integrating  the  curve  with  a  pla- 
nimeter.  However,  the  note  referred  to  is  not  difficult  to 
read,  and  students  are  advised  to  do  so. 

Substituting  the  value  of  I  here  obtained  in  (71),  the 
expression  for  c  is  found  to  be 

c=^,  (73) 

and  is  therefore  proportional  to  h. 


144     THEORY   OF   ERRORS   AND   LEAST   SQUARES 


We  may  now  write  the  error  equation  in  a  final  and  more 
satisfactory  form : 

y^'^e-"'^,  (74) 


in  which  the  law  of  error  distribution  is  made  to  depend 
upon  the  scale-interval  A,  which  is  readily  obtained  from 
the  recorded  observations,  and  upon  a  constant  h  which 
we  have  come  to  refer  to  as  the  precision  index  (Art.  51)  and 
which,  as  will  be  seen  later,  can  also  be  calculated  from  the 
results  of  the  observations. 

55.  Probability  of  an  Error  Lying  between  Given  Limits. 
The  Probability  Integral.  —  An  important  problem  in  the 
theory  of  errors  is  to  find  the  probability  that  an  error 
will  lie  between  two  given  limits  Xi  and  X2.  This  may  be 
obtained  in  terms  of  the  precision  index  h.  For,  the  result 
sought  is  merely  the  sum  of  the  probabilities  of  all  errors 
between  Xi  and  X2,  which  is,  by  (74) 


x  =  Xi 


or  replacing  A  by  dx  as  in  equations  (69),  (70),  in  order  to 
convert  the  summation  approximately  into  an  integral, 

Yi,2  =  -^  r^  e-^'^dx 

=  —  r\-^'-'dx  -  A  T' e-^'-'dx.  (75) 


PRECISION  145 

The  definite  integral  occurring  in  (75),  viz., 

is  simplified  by  a  substitution.     liCt  hx  =  z,  l?x^  =  z^, 
hdx  =  dz ;   when  x  =  X,  z  =  hX.     We  then  have 

F  =  -4  I      e-^'dz  =  -4  I      e-'^'dx.  (76) 

This  expression  Y  is  commonly  called  the  prob- 
ability integral,  and  is  evidently  a  function  of  the 
upper  limit  hX.  It  expresses  the  probability  that 
an  error  will  lie  between  0  and  +  Z.  As  applied 
to  the  question  of  precision,  the  value  of  Y  itself 
is  less  useful  than  that  of  2  7,  which  is  the  proba- 
bility that  an  error  will  lie  between  +  X  and  —  X, 
that  is,  that  a  measured  result  will  be  within  X  of 
the  true  value. 

Our  problem  now  requires  that  we  be  able  to  calculate 
Y  from  the  given  value  oi  hX.  Y  cannot,  however,  be 
expressed  directly  as  a  function  of  hX,  but  must  be  eval- 
uated through  the  use  of  infinite  series.  This  mathemati- 
cal work  is  given  in  Note  D  of  the  Appendix,  to  which  the 
student  is  referred.  Tables  of  the  values  of  the  integral, 
thus  calculated,  are  standard,  and  given  in  every  book  on 
the  theory  of  errors.  In  the  accompanying  table  the 
values  of  2  F  are  given,  corresponding  to  the  argu- 
ment hX. 


146     THEORY   OF   ERRORS   AND   LEAST   SQUARES 


hX 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.0 

0.0000 

0.0113 

0.0226 

0.0338 

0.0451 

0.0564 

0.0676 

0.0789 

0.0901 

0.1013 

0.1 

.1125 

.1236 

.1348 

.1459 

.1569 

.1680 

.1790 

.1900 

.2009 

.2118 

0.2 

.2227 

.2335 

.2443 

.2550 

.2657 

.2763 

.2869 

.2974 

.3079 

.3183 

0.3 

.3286 

.3389 

.3491 

.3593 

.3694 

.3794 

.3893 

.3992 

.4090 

.4187 

0.4 

.4284 

.4380 

.4475 

.4569 

.4662 

.4755 

.4847 

.4937 

.5027 

.5117 

0.5 

0.5205 

0.5292 

0.5379 

0.5465 

0.5549 

0.5633 

0.5716 

0.5798 

0.5879 

0.5959 

0.6 

.6039 

.6117 

.6194 

.6270 

.6346 

.6420 

.6494 

.6566 

.6638 

.6708 

0.7 

.6778 

.6847 

.6914 

.6981 

.7047 

.7112 

.7175 

.7238 

.7300 

.7361 

0.8 

.7421 

.7480 

.7538 

.7595 

.7651 

.7707 

.7761 

.7814 

.7867 

.7918 

0.9 

.7969 

.8019 

.8068 

.8116 

.8163 

.8209 

.8254 

.8299 

.8342 

.8385 

1.0 

0.8427 

0.8468 

0.8508 

0.8548 

0.8586 

0.8624 

0.8661 

0.8698 

0.8733 

0.8768 

1.1 

.8802 

.8835 

.8868 

.8900 

.8931 

.8961 

.8991 

.9020 

.9048 

.9076 

1.2 

.9103 

.9130 

.9155 

.9181 

.9205 

.9229 

.9252 

.9275 

.9297 

.9319 

1.3 

.9340 

.9361 

.9381 

.9400 

.9419 

.9438 

.9456 

.9473 

.9490 

.9507 

1.4 

.9523 

.9539 

.9554 

.9569 

.9583 

.9597 

.9611 

.9624 

.9637 

.9649 

1.5 

0.9661 

0.9673 

0.9684 

0.9695 

0.9706 

0.9716 

0.9726 

0.9736 

0.9745 

0.9755 

1.6 

.9763 

.9772 

.9780 

.9788 

.9796 

.9804 

.9811 

.9818 

.9825 

.9832 

1.7 

.9838 

.9844 

.9850 

.9856 

.9861 

.9867 

.9872 

.9877 

.9882 

.9886 

1.8 

.9891 

.9895 

.9899 

.9903 

.9907 

.9911 

.9915 

.9918 

.9922 

.9925 

1.9 

.9928 

.9931 

.9934 

.9937 

.9939 

.9942 

.9944 

.9947 

.9949 

.9951 

2.0 

0.9953 

0.9955 

0.9957 

0.9959 

0.9961 

0.9963 

0.9964 

0.9966 

0.9967 

0.9969 

2.1 

.9970 

.9972 

.9973 

.9974 

.9975 

.9976 

.9977 

.9979 

.9980 

.9980 

2.2 

.9981 

.9982 

.9983 

.9984 

.9985 

.9985 

.9986 

.9987 

.9987 

.9988 

2.3 

.9989 

.9989 

.9990 

9990 

.9991 

.9991 

.9992 

.9992 

.9992 

.9993 

2.4 

.9993 

.9993 

.9994 

.9994 

.9994 

.9995 

.9995 

.9995 

.9995 

.9996 

2. 

0.9953 

0.9970 

0  9981 

0.9989 

0.9993 

0.9996 

0.9998 

0.9999 

0.9999 

0.9999 

00 

1.0000 

56.  Calculation  of  the  Precision  Index  from  the  Re- 
siduals. —  The  table  in  the  preceding  article  enables  us  to 
find  the  value  of  2  7  corresponding  to  any  given  value  of  X, 
providing  the  precision  index  h  is  known.  That  is,  if  we 
have  h,  we  can  find  from  the  table  the  probability  that  an 
error  will  not  exceed  a  given  value  X  in  numerical  value. 

Reciprocally,  if  the  value  of   2  F  corresponding  to  a 


PRECISION 


147 


given  limiting  error  X  can  be  determined  by  any  means, 
the  same  table  will  give  the  value  of  hX  and  hence  of  h,  just 
as  a  number  can  be  found  from  its  logarithm,  or  an  angle 
from  its  tangent,  by  interpolation.  This  may  be  accom- 
plished from  a  study  of  the  residuals  if  the  number  of 
observations  is  large  enough. 

For,  the  probability  2  Y  that  a  residual  will  lie  between 
+  X  and  —  X  may  be  obtained  by  finding  what  propor- 
tion of  them  do  lie  between  these  limits.  By  choosing 
several  different  values  of  X,  as  many  values  for  h  X  may 
be  found,  which  may  be  combined  like  observation  equa- 
tions for  the  unknown  quantity  h. 

While  this  is  rather  too  laborious  a  method  for  practical 
purposes,  it  will  be  found  a  very  useful  means  of  getting 
clearly  in  mind  the  relation  of  h  to  the  precision  of  the 
measurements.  We  shall  therefore  apply  it,  by  way  of 
illustration,  to  the  following  results  of  144  measurements 
upon  the  length  of  a  line.     Of  these : 


1  gave 

1  gave 

2  gave 

3  gave 

4  gave 

5  gave 
1 1  gave 
16  gave 
21  gave 
16  gave 


2178.1  feet 
.2  feet 
.4  feet 
.5  feet 
.6  feet 
.7  feet 
.8  feet 
.9  feet 

2179.0  feet 
.1  feet 


Residuals 

1.0 
0.9 
0.7 
0.6 
0.5 

■0.4 
0.3 
0.2 

•0.1 
0.0 


18  gave   2179.2  feet 


18  gave 
10  gave 
7  gave 
5  gave 
2  gave 
4  gave 
144 


.3  feet 
.4  feet 
.5  feet 
.6  feet 
.7  feet 
.8  feet 
Mean 
=  2179.1 


Residuals 

+0.1 
+0.2 
+0.3 

+0.4 
+0.5 
+0.6 
+0.7 

=  14.36 


148     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

The  interval  A  between  successive  residuals  is  0.1  foot. 
An  examination  of  the  residuals  gives  the  following 
facts : 


N 

±x 

£Y 

55  are  numerically  not  greater  than 

0.1 

0.3819 

89  are  numerically  not  greater  than 

0.2 

0.6166 

110  are  numerically  not  greater  than 

0.3 

0.7638 

122  are  numerically  not  greater  than 

0.4 

0.8472 

131  are  numerically  not  greater  than 

0.5 

0.9098 

136  are  numeric  illy  not  greater  than 

0.6 

0.9444 

142  are  numerically  not  greater  than 

0.7 

0.9861 

142  are  numerically  not  greater  than 

0.8 

0.9861 

143  are  numerically  not  greater  than 

0.9 

0.9930 

The  numbers  in  the  column  headed  2  Y  are  simply  the 


N  55 

values  of  the  fraction  —  ;  e.g.,  — 

144       ^     144 


0.3819,  etc.    The 


values  of  hX  corresponding  to  these  values  of  2  Y,  ob- 
tained from  the  probability  integral  table,  are  as  follows : 


X 

hX 

X 

hX 

0.1 

0.353 

0.6 

1.353 

0.2 

0.616 

0.7 

1.740 

0.3 

0.838 

0.8 

1.740 

0.4 

1.011 

0.9 

1.907 

0.5 

1.198 

any  one  of  which  will  give  an  approximate  value  of  h. 
By  using  the  above  values,  we  may  write  simple  observa- 
tion equations  for  h,  thus : 


PRECISION  149 

0.1  h  =  0.353, 

0.2  h  =  0.616, 

etc., 

which  when  reduced  in  the  usual  manner  (Art.  34)  give 
as  the  most  probable  value 

h  =  2.33. 

Using  this  as  the  value  of  h  and  0.1  as  that  of  A,  and  sub- 
stituting them  in  (74),  gives  as  the  equation  of  error  distri- 
bution for  this  case 

y  =  0.1314  e-s-*3x«^ 

Let  the  student  plot  this  curve  and  the  actual  distribu- 
tion of  residuals  together  on  the  same  sheet  for  the  purpose 
of  comparison.  The  ordinates  had  better  be  laid  off  on 
five  or  ten  times  as  great  a  scale  as  the  abscissas,  for  con- 
venience. 

57.   Approximate  Formulas  for  the  Precision  Index.  — 

The  foregoing  is  doubtless  as  accurate  a  method  of  ob- 
taining the  precision  index  h  as  could  be  desired  where  the 
number  of  observations  is  large,  and  where  it  can  therefore 
be  assumed  that  the  residuals  distribute  themselves  in 
accordance  with  the  error  law.  It  is  however  too  laborious 
for  practical  purposes,  and  can  be  replaced  by  shorter 
methods.  The  first  one  here  presented  depends,  in  fact, 
upon  the  same  principle  as  that  used  in  the  foregoing  cal- 
culation, being  simply  more  direct. 
Let  there  be  n  observations  made  upon  the  unknown 


150     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

quantity  or  upon  functions  of  it,  the  errors  being  Xi, 
X2,  ' ' ',  Xn-     From  (74),  the  probabiUties  of  these  errors 

are,  respectively,  ,  * 

yi=—ze  ^^ 

Vtt 


^A       _-.2       2 


and  the  probability  of  the  given  set  of  errors  is 

Y  =  (-^k^e-^'^^\  (77) 

The  most  probable  value  of  h  that  can  be  afforded  by 
these  observations  is  the  one  giving  rise  to  the  most 
probable  distribution  of  errors,  a  condition  which  is 
equivalent  to  the  statement  that  Y  is  to  be  a  maximum. 
Hence,  regarding  ^  as  a  variable  and  obtaining  the  condi- 
tion for  Y  a  maximum  by  differentiation  of  (77),  we  have 

whence  ^^\^^'  ^"^^^ 

This  would  be  adequate  if  the  true  errors  x  were  known, 
and  does  very  well  in  any  case,  where  there  are  many  ob- 
servations, if  we  simply  use  the  residuals  p  instead  of  the 
true  errors.    The  discrepancy  between  (78)  and  the  value 


PRECISION  151 

obtained  by  using  the  residuals  is  discussed  by  many 
writers  at  some  length,  and  it  is  a  question  whether  it  de- 
serves such  attention,  inasmuch  as  only  an  approximate 
value  of  h  is  usually  required.  It  may  be  easily  seen  that 
Sx^  is  greater  than  S/a^,  since  by  the  principle  of  least 
squares  2/?^  is  to  be  a  minimum.  Hence  if  S/)^  is  to  be 
used  instead  of  Sa:^  in  (78),  something  must  be  done  to 
reduce  the  numerator  as  well  as  the  denominator.  The 
general  practice  is  to  make  it  n  —  1  instead  of  n,  a  pro- 
cedure which  has  some  theoretical  support.     The  formula 

for  h  now  becomes  , 

^=a/'^-  (79) 

This  formula  is  of  the  greatest  importance  in  the  calcula- 
tion of  what  is  known  as  the  probable  error  (Art.  58).  Its 
use  is  somewhat  laborious,  owing  to  the  necessity  of 
squaring  all  the  residuals.  Another  formula  for  h,  first 
used  by  Peters,  can  be  derived  upon  the  following  reason- 
ing. 

The  total  number  of  errors,  both  +  and  — ,  is  /i.     Then 

if  fix  be  the  number  of  errors  having  the  particular  value  x, 

their  probability  is  , 

2/  =  —  =  -^^"''"'.  (80) 

n      Vtt 

Let  us  consider  only  +  errors,  the  average  value  of  which 
is  the  same  as  that  of  all  the  errors,  +  and  — ,  taken  with- 
out signs.     The  sum  of  all  the  +  errors  is,  from  (80), 


152     THEORY   OF   ERRORS   AND   LEAST   SQUARES 
or  approximately  (see  x\rt.  53), 

The  average  of  the  +  errors,  and  hence  of  all  the  errors 
(disregarding  sign),  is  therefore 

x=oo 


2^  {n^x)        . 


whence  h  =  -^z — .  (81) 

This  formula,  like  (78),  is  in  terms  of  the  errors  x.  In 
order  to  reduce  (78)  to  the  expression  (79)  for  h  in  terms  of 
the  residuals  p,  the  numerator  was  reduced  in  the  ratio 


Vn  —  1  :  Vn. 

If  we  apply  the  same  process  to  (81),  at  the  same  time  re- 
placing the  x's  by  the  /o's,  we  obtain  as  the  analogue  of  (79), 


^^M^:-!)^  (82) 

which  will  be  referred  to  as  Peters'  formula  for  h,  whereas 
(79)  will  be  called  the  standard  formula  for  h. 

It  is  to  be  noted  that  the  above  reasoning  applies  only 
to  observations  of  equal  weight.  The  question  of  weight, 
as  related  to  precision,  will  be  introduced  in  Art.  59. 

58.  The  Probable  Error  of  an  Observation.  —  We  have 
heretofore  treated  the  precision  of  observations  in  a  more 


PRECISION  153 

or  less  abstract  and  relative  way,  and  the  need  is  ap- 
parent for  a  more  concrete  and  tangible  expression  for  it. 
In  short,  we  desire  something  that  will  convey  to  the 
mind  an  idea  of  the  accuracy  attained,  in  terms  of  the 
units  of  measurement  used.  This  has  been  secured  in 
several  ways. 

One  of  the  simplest  quantities  of  this  sort  is  the  average 
residual,  taken  without  reference  to  sign.  Its  relation 
to  the  precision  index  is  obtainable  from  (82),  which 
gives  as  the  average  residual 


2)0^  1    jn-1 


(83) 


(84) 


or  if  n  is  very  large,  approximately, 

2/0^     1 
n       /jVtt* 
which  is  equivalent  to  (81). 

Again,  there  is  the  virtual  or  radical  mean  square  (R.M.S.) 
residual,  which  from  (79)  is 

\   n        h\    2n 
or  if  n  is  large,  approximately, 

J^  =  4-.  (86) 

The  concrete  significance  of  this  quantity  lies  in  the  fact 
that  it  represents  the  abscissa  of  the  point  of  inflection 
on  the  error  curve  (Art.  28,  Eq.  27). 


154     THEORY   OF  ERRORS   AND   LEAST   SQUARES 

The  most  approved  expression  used  for  this  purpose  is, 
however,  the  probable  error.  In  Art.  55  it  is  shown  how  to 
calculate  the  probability  that  an  error  will  not  exceed  a 
given  limit  X,  providing  the  precision  index  h  is  known, 
the  table  of  the  probability  integral  being  used  in  the  cal- 
culation.    We  may  now  give  the  following  definition. 

Designated  by  e,  the  probable  error  of  an  observation  is 
such  that  the  probability  that  the  given  observation  differs 
from  the  truth  by  an  amount  numerically  less  than  e  is 
equal  to  the  probability  that  it  differs  by  an  amount  numeri- 
cally greater  than  e. 

More  briefly,  any  error  is  just  as  likely  to  be  less  than 
the  probable  error  e  as  it  is  to  be  greater ;  or  in  other  words, 
the  probability  that  an  error  lies  between  +  e  and  —  e  is  |. 
In  the  long  run,  half  the  errors  will  lie  within  e,  and  half 
will  exceed  it. 

Therefore,  e  is  that  value  of  the  limit  X,  app)earing  in 
the  a^rgument  hX,  which  corresponds  to  2  F  =  |  =  0.5000 
in  the  table  of  the  probability  integral.  Interpolation 
in  this  table  gives  as  the  value  of  the  argument  hX  for 
which2  y  =  0.5000, 

he  =  0.4769, 


whence  the  probable  error  is 

0.4769 


€  = 


h 

Using  the. standard  formula  (79)  for  A,  this  gives 
e  =  0.6745 


(87) 


J^,  (88) 


PRECISION  155 

in  terms  of  the  sum  of  the  squares  of  the  residuals;  or 
using  Peters'  formula  (82),  we  obtain  Peters'  formula  for 
the  probable  error, 

6  =  0.8453— r=l^=,  (89) 

V7i(n  -  1) 

in  terms  of  the  sum  of  the  residuals  without  sign.  From 
Peters'  formula  it  will  be  seen  that  when  n  is  large,  ap- 
proximately, 

e  =  0.85^;  (90) 

n 

or  the  probable  error  of  an  observation  is  approximately 
equal  to  85  per  cent,  of  the  average  residual,  taken  without 
sign.  This  simple  rule  is  sufficiently  accurate  for  most 
practical  purposes,  in  the  case  of  a  long  series  of  observa- 
tions of  equal  weight. 

The  notation  by  which  probable  errors  are  expressed 
uses  the  double  sign.  For  example,  if  the  mass  of  an 
object,  obtained  by  weighing,  is  stated  as  24.830726 
±  0.000014  grams,  this  means  that  the  probable  error  of 
the  weighing  is  0.000014  gram.  This  quantity  would  be 
obtained,  as  explained  above,  by  taking  a  series  of  weigh- 
ings on  the  same  balance  under  the  same  conditions, 
finding  the  residuals,  and  applying  (88),  (89)  or  (90). 

59.   Relation  between  Probable  Error  and  Weight.  — 

When  the  several  observations  of  the  same  series  are 
assigned  different  weights,  the  probable  error  of  a  single 
observation  has  no  significance  without  further  qualifica- 
tion, since  the  precision  index  h,  and  hence  the  probable 


156     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

error,  is  supposed  different  for  the  different  observations. 
We  may  express  both,  however,  with  reference  to  obser- 
vations of  some  selected  precision,  as  those  which  have 
been  assigned  unit  weight.  We  shall  designate  by  h  the 
precision  index  and  by  e  the  probable  error  of  observations 
of  unit  weight,  and  refer  to  them  also  as  observations  of 
standard  precision.  Other  observations  whose  weights 
are  Wi,  Wi,  •••,  have  precision  indices  ^i,  fe,  ••*,  and 
probable  errors  ei,  €2,  •••• 

We  have  already  deduced  the  relation  between  h  and  w 
(Art.  51,  Eq.  65).     It  is 

h:  h:  hz: '"  =^wi:  ^W2:  ^wz'.  "', 

From  (87),     /^i :  fe  :  ^3  :-=-:- :  -:•••. 

ei     €2     es 

Combining  these  proportions, 

€1:  €2:  €3:  •••  =-=  .  -=.  -=:  •••,     (91; 

"^Wi        ^W2       ^Wz 

which  expresses  the  very  important  principle  that  the 
'probable  errors  of  different  observations  in  the  same  series 
are  inversely  proportional  to  the  square  roots  of  their  weights; 
or  reciprocally,  the  iceiyhts  of  observations  of  the  same  kind 
are  inversely  proportional  to  the  squares  of  their  probable 
errors. 

To  illustrate  this,  if  the  probable  error  of  a  measured 
quantity  obtained  by  one  method  is  found  to  be  only 
one-half  as  great  as  that  obtained  by  a  less  precise  method, 
then  the  weight  assigned  to  the  former  in  combining  them 
should  be  four  times  that  assigned  to  the  latter.     In  other 


PRECISION  157 

words,  one  observation  by  the  former  method  is  worth 
four  made  by  the  latter. 

If  €  be  the  probable  error  of  an  observation  of  unit 
weight,  found  from  (88),  (89)  or  (90),  then  by  the  foregoing 
principle,  the  probable  error  of  an  observation  of  weight  w 
is  given  by 

6,  =  ^.  (92) 

This  will  shortly  be  seen  to  have  an  important  applica- 
tion to  the  finding  of  probable  errors  of  adjusted  or  most 
probable  values  of  unknown  quantities. 

If  the  probable  error  of  an  observation  of  unit  weight 
is  to  be  calculated  from  a  series  of  weighted  observations, 
we  may  generalize  the  reasoning  of  Art.  57  as  follows. 
The  weights  are  Wi,  w^,  •••,  and  the  corresponding  precision 
indices  h,  h,'"  -     h  being  the  standard  index, 

hi  =  h^wu 
hi  =  h^W2y 

hn  =  h^Wn. 
The  probabilities  of  the  respective  errors  are  now  given  by 


A 


y2  =  -=^h< 


v: 


w-ie 


-h^WlX^ 


158      THEORY   OF   ERRORS   AND   LEAST   SQUARES 
The  joint  probability,  corresponding  to  (77),  is  now 

from  which,  by  the  same  reasoning  as  that  leading  to  (78) 
and  (79),  the  standard  index  of  precision  is 


-V 


2S(«,/>2)-  ^^^^ 


Instead  of  (88)  we  now  have,  by  substitution  of  this  new 
expression  for  h  in  (87), 


a/???. 


€  =  0.6745.  p^^^,  (94) 

\  n  —  1 

the  important  standard  formula  for  the  probable  error  of 
an  observation  of  unit  weight,  as  obtained  from  a  series 
of  weighted  observations.  In  this  formula,  before  sum- 
ming the  squares  of  the  residuals,  each  square  is  multi- 
plied by  the  corresponding  weight;  or,  otherwise,  each 
residual  is  multiplied  by  the  square  root  of  the  corre- 
sponding weight.     (See  Art.  52.) 

The  same   modification  may  be  made  in  the  Peters' 
formula  (89)  to  adapt  it  to  weighted  observations,  giving 

*  =  0.8453^^^L,  (95) 

Vn(n  —  1) 

or  if  w  is  large,  approximately, 

,  =  0.85?^^,  (96) 

n 

which  corresponds  to  (90). 


PRECISION  159 

EXERCISES 

60.  1.  Two  specific  gravity  bottles,  one  of  which,  No. 
7701  a,  was  of  the  ordinary  type,  and  the  other.  No.  7701  c, 
of  a  special  improved  design,  were  each  filled  with  water 
five  times  at  the  same  temperature,  the  following  being 
the  results  of  the  weighings,  which  were  made  on  the  same 
balance  in  the  same  manner : 

No.  7701  a  No.  7701  c 

42.602818  45.345518 

42.604108  45.345852 

42.603512  45.345597 

42.602062  45.346437 

42.602947  45.346219 

Find  the  probable  error  of  a  single  filling  and  weighing 
with  each  of  the  two  bottles,  and  the  relative  weights  of 
a  single  observation  in  the  two  cases. 

2.  Eighteen  measures  of  a  horizontal  angle  were  made 
by  means  of  a  large  Coast  Survey  theodolite,  as  follows, 
the  observations  being  of  equal  weight : 

13°  31'  17''.6  13°  31'  20''.4 

21    .5  20   .9 


19  .0 

23  .5 

21  .5 

18  .4 

26  .2 

14  .2 

17  .1 

21  .0 

22  .1 

21  .8 

20  .1 

22  .4 

17  .9 

17  .6 

160  THEORY   OF   ERRORS   AND    LEAST   SQUARES 

Find  the  probable  error  of  a  single  observation  of  this 
series  by  means  of  each  of  the  formulas  (88),  (89)  and  (90). 
Regarding  the  mean  as  an  observation  of  weight  18, 
find  the  probable  error  of  the  mean. 

3.  Find  the  probable  error  of  one  shot  in  your  own  target 
experiment  of  Art.  11,  Ex.  1. 

4.  Find  the  probable  error  of  one  observation  in  the 
series  of  measurements  which  you  made  upon  a  line  in 
Art.  11,  Ex.  2.     Also,  find  the  probable  error  of  the  mean. 

6.  Six  separate  researches,  by  different  observers,  upon 
the  velocity  of  light  gave  the  following  mean  results,  with 
their  probable  errors,  in  kilometers  per  second : 

298000  ±  1000 
298500  ±  1000 
299930  ±  100 
299990  ±  200 
300100  ±  1000 
299944  ±   50 

Assign  the  proper  relative  weights  and  find  the  probable 
error  of  an  observation  of  unit  weight. 

Also,  regarding  the  weighted  mean  as  an  observation 
of  weight  Ziv,  find  its  probable  error. 

Explain  why  the  answer  to  the  first  part  of  the  problem 
is  not  1000,  supposing  the  first  observation  to  be  assigned 
unit  weight.  From  the  answer  to  the  second  part,  do  the 
less  precise  observations  add  to  the  value  of  the  whole? 
Give  reason  for  your  conclusion. 


PRECISION 


161 


6.  The  constant  of  a  Babinet  compensator  is  determined 
by  measuring  the  distance  between  two  successive  dark 
bands  as  seen  through  the  analyzer.  Micrometer  readings 
were  taken  as  follows : 


IsT  Band 

2d  Band 

IsT  Band 

2d  Band 

267 

225 

267 

225 

269 

224 

265 

227 

268 

226 

268 

223 

267 

227 

267 

227 

264 

226 

264 

226 

266 

226 

266 

225 

266 

227 

264 

227 

268 

225 

267 

226 

268 

224 

266 

224 

264 

225 

267 

226 

Find  the  probable  error  of  one  measurement  of  the  differ- 
ence in  readings;   of  the  mean. 

7.  Ten  measurements  were  made  upon  the  magnitude 
of  a  certain  bright  star,  with  the  following  results : 

0.600  0.470 

.460  .483 

.477  .475 

.500  .490 

.467  .475 

Find  the  probable  error  of  one  measurement  and  of  the 
mean. 

8.  Syntheses  of  carbonic  acid  gas  made  from  different 
kinds  of  carbon  by  Dumas  and  Stas  gave  the  following 


162      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

results    (Freund,    Cheinical   Composition).     The   numbers 
represent  the  percentage  of  carbon  in  the  gas. 


Natural  Graphite 

Artificial  Graphite 

Diamond 

27.241 

.268 
.270 

.258 
.248 

27.237 
.253 
.281 
.307 

27.251 
.276 
.301 
.263 
.275 

Find  the  probable  error  of  one  determination  and  of  the 
mean. 

9.  In  a  series  of  base  line  measurements  made  with 
both  steel  and  invar  tapes,  the  following  probable  errors 
were  found  (U.  S.  Coast  Survey  Report,  1907) : 


Base  Line 

Steel 

Invar 

Point  Isabel 

Willamette 

Tacoma 

Stephen      

Brown  Valley      .... 
Royalton 

ONE    PART  IN 

1   300   000 
1    730   000 
1    630   000 
1    120  000 

1  420   000 

2  260   000 

ONE    PART   IN 

2  310   000 

3  340  000 
2   980   000 

2  040   000 

3  110   000 
2   460   000 

Averaging  these,  find  the  relative  weights  of  base  line 
measurements  made  with  these  two  tapes. 

10.  Apply  Peters'  formula  (95)  to  find  the  probable 
error  of  an  observation  of  unit  weight  for  the  data  of  Ex. 
14,  Art.  49. 


PRECISION  163 

61.  Probable  Errors  of  Functions  of  Observed  Quanti- 
ties. —  An  important  phase  of  the  subject  of  preci- 
sion is  what  may  be  termed  the  "propagation  of 
error "  and  illustrated  by  an  example :  The  prob- 
able error  of  the  diameter  of  a  circle,  obtained  by 
measurement,  is  c;  what  is  the  probable  error  E  of 
the  area  calculated  therefrom?  Or  generally,  given  the 
probable  error  of  a  measured  value  of  a  quantity,  to 
find  the  corresponding  probable  error  of  any  function  of 
that  quantity. 

Let  the  measured  quantity  be  g,  and  the  function 

e=/(9). 

Let  an  observation  be  made  upon  q  with  error  x^  and  let 
the  corresponding  error  affecting  the  function  Q,  as  a 
result  of  this,  be  X.  Then  if  x  be  small,  we  have  ap- 
proximately  X:x  =  dq:dq, 

or  X='^fx. 

dq 

It  may  now  readily  be  seen  that  if  e  and  E  are  the  prob- 
able errors  of  the  measured  q  and  of  Q,  respectively,  then 

E=  ^€.  (97) 

dq 

This  may,  however,  be  shown  as  follows. 

If  Xi,  a'2,  •••,  Xn  are  a  series  of  errors  committed  in 
measurements  upon  q,  and  Xi,  X2,  •••,  Xn  are  the  resulting 
errors  in  Q,  then  as  above. 


164  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

dq 
Y      dQ 


X  =^X 

"       dq   ' 


2Z2  =  ('^Ysx2.  (98) 


or  squaring  and  adding, 

dq. 

Now  from  (87),  substituting  the  value  of  h  given  in  (78), 
since  the  ar's  and  the  Z's  are  true  errors  (not  residuals), 
the  probable  error  of  ^  is  

€  =0.6745  \/—, 


and  that  of  Q, 

E  =  0.6745  y 

from  which 

^^2   _        e'^ 

0.67452' 

^Y^=     ^^n 

/SZ2 


0.67452 

The  substitution  of  these  in  (98)  with  subsequent  reduction 
gives  (97). 

That  is,  the  probable  error  of  a  function  of  a  single  measured 
quantity  is  equal  to  the  derivative  of  the  function  times  the 
probable  error  of  the  measured  quantity. 

For  example,  if  the  measured  radius  of  a  circle  be 
^  =  9,67  ±  0.02   cm.,  the  computed  area   is  Q  =  Tq^  = 


PRECISION  165 

293.7663  sq.  cm.,  and  its  probable  error  is  E  =  ±  2  7rg  X 
0.02  =  ±  1.215  sq.  cm. 

In  general,  Q  is  a  function  of  several  (/)  measured  quan- 
tides:  «=/(?!,  <?2, -,  9,).  (99) 

Then  if  Xi,  X2,  -",  Xi  are  the  errors  of  the  respective  values 
of  qi,  ^2,  '",  Qi  simultaneously  substituted  in  (99),  the 
resulting  error  of  Q  is  given  approximately  by 

X  =  |^x,+|2x.+  ...  +  |«x,.  (100) 

Let  there  be  a  number  (n)  of  series  of  observations  upon 
the  ^'s,  each  giving  rise  to  an  error  X  as  represented  in 
(100),  viz.,  Xi,  X2,  ••♦,  X„.     Then  approximately, 

which  is  obtained  from  the  X's  upon  squaring  (100)  and 
omitting  the  product  terms  of  the  expansion.  This  omis- 
sion is  justified  by  the  fact  that  there  will  be  in  the  long 
run  as  many  +  products  as  — ,  and  they  will  be  distrib- 
uted approximately  in  accordance  with  the  error  law, 
and  will  hence  practically  cancel  each  other;  whereas, 
the  square  terms  are  all  +,  and  must  be  retained. 

By  the  same  reasoning  as  that  employed  in  the  simpler 
case  following  (98),  we  now  readily  obtain 

of  which  (97)  may  be  regarded  as  a  special  case.  The 
quantities  ei,  €2,  ••♦,  e^  are  the  probable  errors  of  measured 


166  THEORY   OF   ERRORS   AND   LEAST   SQUARES 

values  of  qi,  qi,  •••,  qi,  and  E  the  probable  error  of  Q  re- 
sulting from  substituting  these  values  in  the  function  (99). 
As  special  cases  of  importance,  we  may  take  the  following : 

(a)      If  q  =  Kiqi  +  K2q2-^"'+Kiq„ 


then  E  =  ^KiW  +  K^W  +  •  •  •  -^Ki\\      (103) 

(6)      If  Q=Kqi^q^'"q,\ 

'^"  -=#?)'-.'^'^)"-=--^-+(f)'-.--  («) 

Let  the  student  deduce  these  results. 


62.  Probable  Errors  of  Adjusted  Values.  —  The  dis- 
cussions of  the  probable  error  heretofore  have  been  con- 
fined to  the  results  of  single  measurements.  The  values 
finally  taken  as  the  most  probable,  for  the  unknown  quan- 
tities, from  a  series  of  measurements  may,  however,  be  more 
trustworthy  than  that  of  any  single  measurement,  and  the 
manner  of  their  calculation  from  the  observations  enables 
us,  by  applying  the  laws  developed  in  the  preceding  article, 
to  calculate  the  probable  errors  of  these  adjusted  values, 
regarding  them  as  functions  of  the  observations. 

As  the  simplest  illustration,  we  shall  first  take  the  case 
of  direct  observations  of  equal  weight  upon  a  single  quan- 
tity q.    The  observation  equations  are 

q  =  5i, 

q     =     S2y 


PRECISION  167 

there  being  n  observations.  The  most  probable  value  m 
may  be  given  in  the  form 

n         n  n 

This  is  a  function  of  the  form  (a)  of  the  preceding  article, 
and  the  probable  error  is  given  by  (103).  Each  observa- 
tion s  has  the  same  probable  error,  designated  by  (88), 


€  =  0.6745 


Then  by  (103)  the  probable  error  of  the  arithmetical 
mean  m  is 

e„  =  J4  +  4+  -  +^  =  -^  =  0.6745   /-^, (105) 

which  is  the  formula  ordinarily  applied.  It  may  be  ob- 
tained at  once  from  (88)  and  (92)  by  regarding  the  arith- 
metical mean  of  n  observations  of  unit  weight  as  an 
observation  of  weight  ri,  as  suggested  in  certain  of  the 
problems  of  Art.  60. 

By  a  similar  course  of  reasoning,  if  there  are  n  observa- 
tions upon  a  single  quantity  having  weights  Wi,W2,  •••,  iVn 
assigned,  the  probable  error  of  the  weighted  mean  (Art.  48) 

e„„  =  0.6745,/^^^.  (106) 

\  (n  —  l)Xw 

The  general  case,  in  which  there  are  n  observations  of 
different  weight  upon  functions  of  /  unknowns  qi,  q^,  •••, 
qi,  is  somewhat  more  complicated.*     We  shall  deal  only 

*  Sae  article  by  the  author,  Popular  Astronomy,  Vol.  XIX, 
p  239. 


168      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

with  the  usual  problem  of  first-degree  observation  equa- 
tions, represented  by  (38),  Art.  34.  The  residuals  are 
then  given  by  (39),  from  which  their  numerical  values 
must  first  be  calculated.  The  probable  error  of  an  ob- 
servation of  unit  weight  is  now  calculated  from  these 
residuals  in  the  usual  manner,  or  by  another  formula 


'V^. 


€  =  0.6745.  r^^^^,  (107) 

\    n  —  I 

which  is  commonly  taken  as  being  more  satisfactory,  and 
which  certainly  differs  little  from  (94)  when  n  is,  as  it 
should  be,  large  compared  with  /.  (94)  may  be  regarded 
as  a  special  case  of  (107),  in  which  /  =  1. 

In  many  kinds  of  work,  the  probable  error  of  an  ob- 
servation of  unit  weight  is  known  to  the  observer  through 
long  experience  with  his  instruments,  and  need  not  be 
calculated  with  reference  to  each  series  adjusted.  At 
any  rate,  we  shall  suppose  the  probable  errors  of  ^i,  S2,  •••, 
Sn  in  equations  (38)  to  be  known,  and  designate  them  by 
^1,  ^2,  ••*,  ^n-  It  is  now  required  to  find  the  probable 
errors  of  mi,  m2,   •••,  m^,  which  may  be  designated  by 

€l,    €2,  •",    €;. 

In  the  process  of  adjustment  of  such  a  set  of  observa- 
tions as  here  referred  to,  there  arises  a  set  of  /  most  prob- 
able or  normal  equations,  which,  from  the  mode  of  arriving 
at  them  (Arts.  34  and  48) ,  may  be  symbolized  as  follows : 

^{aws)  —  {Aivii  +  Biiih  +  •  •  •  +  Riiih)  =  0, 

etc.,  or  more  conveniently, 


PRECISION 


169 


^{bws)  =  A2ini  +  B2'm2  +  •••  +  Ri^ih 


(108) 


^(rws)  =  AiTTii  +  Bim2  +  •  •  •  +  Rinii.  . 

If  we  represent  by  Ei,  E2,  -",  E^  the  probable  errors 
of  the  members  of  these  respective  normal  equations, 
then  these  E's  may  each  be  expressed  in  two  ways.  In 
the  first  place,  since  the  first  member  of  the  first  normal 
equation  is 

^{aws)  =  aiWiSi  +  a2W2S2  +  •••  +  anWnSn, 

its  probable  error  Ei,  as  a  function  of  the  s's,  is  given  by 

El'  =  ai'wiW+a2'w2W+'  •  •  ^an'wnV  =  ^{aVe''),     (109) 

with  similar  relations  for  the  other  normal  equations. 
Again,  the  probable  error  Ei  of  the  second  member  of  the 
first  normal  equation  (108),  as  a  function  of  the  m's,  is 
given  by 


^iV  +  5iV4--+i?iV, 


(110) 


with  similar  relations  for  E2,  Es,  ••♦,  Ei.  Then  equating 
(109)  and  (110),  and  the  other  similar  pairs,  we  obtain 
the  system 

^iV  +  Bi'e,^  +"'+  R,W  =  ^{aWe'),  ' 


A2W  +  B2W+  -  +R2W=^(bVe'), 


AiW-\-  5,V+  •••  +  RiW=^(rVe'). 


(Ill) 


These  equations  are  of  the  first  degree  in  et^,  €2-,  •••,  e^^ 
and  may  be  readily  solved  for  these  values,  the  required 


170      THEORY   OF   ERRORS   AND   LEAST   SQUARES 

probable  errors  themselves  being  then  obtained  by  ex- 
tracting the  square  roots. 

The  actual  process  is  often  not  as  complicated  as  might 
be  supposed,  especially  when  the  number  of  unknowns  is 
not  large.  The  coefficients  Aj  B,  •••,  R  appearing  in  (111) 
are  already  known  from  the  normal  equations.  The 
weights  w  are  simple  numbers,  frequently  all  unity.  The 
work  is  greatly  facilitated  by  the  use  of  tables  of  squares 
and  square  roots,  and  the  slide  rule.  And  it  may  finally 
be  remarked  that,  since  no  great  precision  is  required  in  de- 
termining probable  errors,  superfluous  decimal  places  may 
be  dispensed  with  in  the  several  stages  of  their  calculation. 

Another  method  of  calculating  the  probable  errors  of 
adjusted  values  from  observation  equations  of  the  first 
degree  is  given,  without  proof,  in  the  Appendix,  Note  E. 

63.  Probable  Errors  of  Conditioned  Observations.  — 
The  laws  developed  in  Art.  61  make  possible  the  extension 
of  the  foregoing  processes  to  the  case  of  observations  upon 
quantities  limited  by  rigorous  conditions  (Art.  40).  It 
will  be  remembered '  that  the  m  equations  of  condition 
are  first  used  to  obtain  the  value  of  m  of  the  unknowns 
in  terms  of  the  others,  those  values  being  substituted  for 
them  in  the  observation  equations  before  adjustment. 
The  probable  errors  of  the  quantities  still  involved  in  the 
observation  equations  may  now  be  found  as  explained 
in  the  preceding  article.  This  being  done,  the  probable 
errors  of  the  m  eliminated  quantities  may  be  found  as 
functions  of  the  others,  by  means  of  (102). 


PRECISION  171 

EXERCISES 

64.    1.    (The  formula  for  double  weighing  on  a  balance 

is  W  =  p  -{-  -^— — -,  in  which  p  is  the  sum  of  the  weights 

used,  ri  and  r2  are  the  pointer  readings  when  object  is  on 
left  and  right  pans,  respectively,  and  s  is  the  sensibility 
of  the  balance.)  For  ten  weighings  of  the  same  object, 
the  values  of  n  —  r^  were  as  follows : 


0.96 

0.93 

1.08 

0.95 

0.99 

1.12 

1.02 

1.05 

0.92 

1.10 

The  factor  —  for  the  load  used  was  0.0002753.     Find 
2s 

the  probable  error  of  one  weighing,  and  of  the  mean  of  the 
ten  weighings.    (Does  p  need  to  be  given  for  this  purpose  ?) 

2.  Given,  the  probable  errors  of  two  measured  quanti- 
ties, to  find  the  probable  error  of  their  calculated  sum  or 
difference. 

3.  All  of  the  weighings,  the  data  for  which  are  given 
in  Ex.  1,  Art.  60,  were  made  on  the  same  balance  and  by 
the  same  method  as  those  giving  rise  to  the  data  of  Ex.  1 
of  this  article.  The  latter  data  refer  to  ten  weighings  of 
the  empty  bottle  No.  7701  a,  for  which  p  =  17.423  g. 
The  former  data,  referring  to  bottle  No.  7701  a,  are  for 
fillings  with  pure  water  at  21°  C,  at  which  temperature 


172     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

the  specific  volume  of  water  is  1.001957  cc.  per  gram. 
Find  the  most  probable  capacity  of  the  bottle  at  this 
temperature,  and  the  probable  error  of  the  result. 

Find  the  probable  error  of  a  single  determination  of  the 
capacity,  the  above  data  being  regarded  as  five  deter- 
minations. 

4.  Given,  the  probable  errors  of  the  measured  legs  of  a 
right  triangle,  to  find  the  probable  error  of  the  calculated 
hypothenuse. 

6.  Given,  the  probable  error  of  a  measured  angle,  to 
find  the  probable  error  of  its  sine,  derived  therefrom; 
of  its  tangent. 

6.  Find  the  probable  errors  of  the  constants  Vo  and  K 
calculated  from  the  data  of  illustration  2,  Art.  36  (p.  80). 

7.  Find  the  probable  errors  of  the  constants  a,  6,  c 
calculated  from  the  data  of  illustration  1,  Art.  44  (p.  107). 

8.  (From  J.  P.  Bartlett,  Least  Squares.)  The  following 
weighted  observations  were  made  upon  the  differences  of 
longitude  of  four  American  observatories: 

Cambridge  —  Washington  23  m.  41.041  s.,  wt.  30 
Cambridge  —  Cleveland  42  m.  14.875  s.,  wt.  7 
Cambridge  —  Columbus  47  m.  27.713  s.,  wt.  8 
Washington  —  Columbus  23  m.  46.816  s.,  wt.  7 
Cleveland  —  Columbus         5  m.  12.929  s.,  wt.    5 

The  longitude  of  Washington  being  taken  as  5  h.  8  m. 
15.78  s.,  find  the  most  probable  longitude  of  each  of  the 
other  stations,  with  their  respective  probable  errors. 


PRECISION 


173 


9.  The  probable  error  of  a  single  observation  upon  an 
angle  with  a  surveyor's  transit  is  known  to  be  ±  1'  4''. 
If  the  angle  ^  of  a  triangle  is  measured  three  times, 
B  five  times  and  C  six  times  with  this  instrument,  calcu- 
late the  probable  errors  of  the  most  probable  values  of 
A,  B,  C,  obtained  by  adjusting  these  measurements. 

10.  Adapt  Peters'  formulas  to  the  probable  errors  of 
adjusted  values,  (105),  (106);  also  to  (107). 

11.  If  the  current  in  a  galvanometer  corresponding  to 
deflection  5  is  c  =  K  tan  8,  find  the  probable  error  of  a 
current  determined  from  a  deflection  reading  whose  prob- 
able error  is  4'. 

12.  The  following  table  occurs  in  a  certain  Coast  Survey 
report,  referring  to  the  probable  errors  of  the  various  sec- 
tions of  a  base  line,  in  millimeters. 


Sec. 

P.  E.  Due  to 

Uncertainty  in 

Lengths  of 

Tapes 

Due  to 

Uncertainty  in 

Coefficients  of 

Exp. 

Due  to 

Accidental 

Errors  of 

Meas. 

Combined 

P.  E.  of  Each 

Section 

1 

±0.21 

±0.05 

±1.75 

± 

2 

.21 

.06 

0.71 

3 

.21 

.01 

0.54 

4 

.21 

.02 

0.24 

5 

.22 

.09 

1.72 

6 

.22 

.09 

0.54 

7 

.22 

.04 

2.06 

8 

.22 

.13 

2.53 

9 

.21 

.15 

0.30 

10 

.21 

.11 

1.08 

11 

.21 

.09 

1.75 

12 

.21 

.08 

0.20 

13 

.21 

.12 

1.92 

14 

.03 

.02 

0.07 

174     THEORY   OF   ERRORS   AND   LEAST   SQUARES 

Fill  out  the  last  column  and  obtain  the  probable  error  of 
the  whole  base  line. 

13.  The  most  probable  length  of  the  Stanton  Base  is 
13191.3417  ±  0.0052  meters.  Find  the  most  probable 
value  and  probable  error  of  its  logarithm.  (Omit  unless 
at  least  a  seven-place  table  is  available.) 

14.  The  weights  of  three  measured  angles,  BAC,  CAD, 
DAEf  are  2,  1,  5,  respectively.  Find  the  corresponding 
weight  of  the  angle  BAE  obtained  by  adding  the  measure- 
ments. 

15.  The  probable  error  of  a  circle  reading  on  a  transit 
is  0'.2,  and  of  a  pointing  at  a  signal,  OM.  What  is  the 
probable  error  of  a  single  differential  angle  measurement  ? 

16.  The  probable  error  of  a  setting  on  a  mark  being 
€i,  and  of  a  circle  reading,  €2,  find  the  probable  error  of  an 
angle  measurement  by  the  cumulative  method,  using  n 
turns  of  the  circle. 

17.  The  probable  error  of  a  scale  reading  on  a  cathetom- 
eter  is  0.07  mm. ;  of  a  setting  of  the  telescope  on  a  mark, 
O'.l ;  and  of  an  adjustment  of  the  level,  0'.07.  Find  the 
probable  error  of  the  mean  of  ten  readings  on  a  mark  2 
meters  from  the  instrument.  If  you  had  to  use  a  cathe- 
tometer,  would  you  analyze  the  probable  error  in  this 
way  ?     How  would  you  find  it  ? 

18.  Following  are  the  results  of  three  series  of  meas- 
urements on  the  combining  weight  of  lithium,  made  by 
different  chemists  (Freund,  Chemical  Composition) : 


PRECISION 

Diehl  .     . 

,     .     59.417  ±  0.0060 

Troost      . 

.     .     59.456  ±  0.0200 

Dittmar  . 

.     .     59.638  ±  0.0173 

175 


Weight  these  results  and  obtain  the  weighted  mean  and 
its  probable  error. 

19.  Five  independent  series  of  determinations  of  the 
atomic  weight  of  silver  gave  the  following  results  (Freund, 
Chemical  Composition) : . 

107.9401  ±  0.0058 
107.9406  ±  0.0049 
107.9233  ±  0.0140 
107.9371  ±  0.0045 
107.9270  ±  0.0090 

Assign  weights  and  obtain  the  weighted  mean  and  its 
probable  error. 

What  would  be  the  effect  of  a  persistent  error  entering 
one  of  such  a  series?  Might  the  probable  error  of  a 
weighted  mean  ever  be  greater  than  that  of  any  one  of  the 
observations  entering  into  it  ? 

20.  Apply  the  appropriate  Peters'  formula  to  finding  the 
probable  error  of  the  weighted  mean  of  the  observations 
of  Ex.  14,  Art.  49.     (See  Ex.  10,  Art.  60.) 

21.  The  probable  error  of  the  mean  of  fifty  observa- 
tions is  found  to  be  0.1  per  cent.  How  many  more  ob- 
servations would  be  necessary  to  reduce  it  to  0.01  per 
cent.  ? 


APPENDIX 

SUPPLEMENTARY   NOTES 

A.  Proof  of  the  Necessary  Functional  Relation  As- 
sumed in  Deriving  the  Error  Law.  (Supplementary  to 
Art.  27.)  —  To  deduce  the  form  of  the  function  4>,  such 
that  any  set  of  values  of  Xi,  X2,  ...,Xn  that  will  render 

X  =  xi-\-X2+  '"  -\-Xn  =  ^  (a) 

will  simultaneously  render 

^  =  <^fe)  +  <^fe)+  -  +c^(a:J  =0.  (6) 

Let  us  add  a  small  finite  quantity  e  to  any  one  of  the 
oj's,  say  Xr,  and  subtract  it  from  any  other,  say  Xg,  making 
the  new  values  of  these  quantities  Xr  =  Xr  -\-  ^y  Xa  —  Xa 
—  e.  This  will  not  alter  the  condition  X  =  0,  and  hence 
will  not  alter  the  condition  <I>  =  0,  since,  by  the  hypothe- 
sis, these  conditions  are  to  be  simultaneous.  This  necessi- 
tates that    ^/    X  ,    ^/    X       j_r       \     \  \   ^r  \ 

^'  *^^^      lct>(Xr  +  €)  -  <t>{Xr)]  +  [cf>iXs  "  e)  -  <f>ix,)]  =  0, 

whatever  the  values  of  Xr,  Xa  and  e. 

Dividing  through  by  e,  this  may  be  written 

(l>(Xr  +  €)  -  <t>{Xr)  _  <t)(Xs  -  c)  -  0fe)  ^  q 

e  — e 

N  177 


178  APPENDIX 

Allowing  e  to  approach  zero,  this  becomes  at  the  limit 

— -<^(a-,)-— -(</)X,)  =  0; 
axr  aXs 

or  since  Xr  and  Xa  represent  any  two  of  the  x^s,  in  general, 

—-<f>{xi)=-—<t>(x2)=  •••  =3— 0(a;J. 
dxi  ax2  dXn 

It  follows  at  once  that,  since  the  x's  may  be  varied  in 
any  manner  among  themselves,  only  so  condition  (a)  holds, 

—  (f){x)  is  a  constant,  say  K.     Therefore,  integrating, 
ax 

(f){x)  =  Kx  +  c. 

That  c  =  0  follows  from  (a)  and  (6)  jointly,  since  substi- 
tution in  (6)  gives 

^  =  K(xi  -\-X2  +  "'  +Xn)  +nc  =  0, 

the  first  term  of  which  vanishes  by  (a). 

Hence,  necessarily,       (f)(x)  =  Kx, 
which  is  Eq.  (22),  Art.  27. 

B.  Approximation  Method  for  Observation  Equations 
Not  of  the  First  Degree.  (Supplementary  to  Art.  39.)  — 
This  method  requires  that  the  values  of  the  unknowns  be 
very  approximately  known  beforehand,  as  by  choosing 
such  of  the  observation  equations  as,  when  solved  simul- 
taneously, will  yield  values  for  all  of  them.  Attention  is 
then  given  to  the  unknown  small  corrections  that  must  be 
applied  to  these  approximate  values;  a  procedure  some- 


APPENDIX  179 

what  resembling  Horner's  method  of  approximation  for 
algebraic  equations. 

The  approximate  values,  however  obtained,  being  des- 
ignated by  ai,  a2,  ••*,  a^,  and  the  corrections  required  by 
q\y  q'2,  •",  q\,  the  true  values  of  the  unknowns  are 

qi  =  ai  +  q\, ' 
92  =  a2  +  g'2, 

qi  =  G.i  +  q'l- . 

Let  the  non-linear  observation  equations  to  be  dealt  with 
betypifiedby         ;(,^^  ,^,  ...,,,)  =,  (^) 

The  substitution  of  the  values  (c)  in  ((i)  gives 

/(ai  +  q\,  a2  +  q'l,  •••,  a^  +  4i)  =  ^,  W 

an  observation  equation  in  which  the  unknowns  are  the 
small  corrections  q\y  q'2,  -")  q\.  Expanding  the  first 
member  of  this  by  the  general  Taylor's  theorem, 

f{o.\-\-q'h  ai-^-q'-i,    ",  0.1  + q'l)  =/(ai,  a2,  •••,  a^) 

+  9'i— /(ai,  a2,  •••,  ai)+g'2  — /(ai,  a2,  •••,  a^)  +  ••• 
oai  oa2 

+  q'l—fi^h  a2,  •••,  a^)  +  R, 
aai 

R  being  the  remainder  of  the  series,  which  involves 
higher  powers  of  the  very  small  corrections  q'l,  etc., 
and  which  may  therefore  he  neglected  without  serious 
inaccuracy. 


180 

APPENDIX 

Denoting 

fiai; 

,   (12,    ' 

",a,)hyF, 

dF 

5ai 

by  a, 

dF 

5a2 

hyh, 

oa.1 

we  may  therefore  replace  {e)  by 

or  ag'i  +  &g'2+-+V,  =  /,  (/) 

in  which  s'  denotes  s  —  F. 

This  is  an  observation  equation  of  the  first  degree,  and 
may  be  used  as  such,  in  combination  with  other  observa- 
tions similarly  obtained  from  their  respective  originals, 
for  finding  the  most  probable  values  of  the  corrections. 
This  being  done,  the  most  probable  values  of  the  unknown 
quantities  themselves  are  found  by  adding  the  most 
probable  corrections  to  the  approximate  values  a. 

C.  Evaluation  of  the  Integral  f  e-^'^^'dx.  (Supple- 
mentary to  Art.  54.)  —  Equation  (70)  is 

I  =JJe-^'^'dx. 

This  may  be  transformed  into 


APPENDIX  181 

The  new  integral  V  is   independent  of    h.     For,  let 
hx  =  z,  then  hdx  —  dz,  and 

/'=  Ce-''dz=  Ce-'^'dx.  (h) 

Returning,  however,  to  the  original  form  of  J'  in  (g), 
multiply  it  by  e~^^dh : 

I'e-^'dh  =  r"[g-(i+-^)'^^  .  hdh]dx. 

Jo  * 

r  and  X  being  independent  of  /j,  we  may  integrate  both 
members  of  this  equation  with  respect  to  h  as  a  parameter, 
assigning  the  limits  0  and  oo   to  this  integration  also, 

tnUS  I  /»G0  /»x  r  /"x  ~| 

rj    e-^dh=j      J    e-^'^^'^^'-hdh\dx.  (i) 

Now  the  integral  within  the  brackets  is  readily  de- 
termined : 

r^-a+x^)h^ .  hdh  = ^—^  Pg-(i+x^)A^ .  2(1 -^x^)hdh 

Jo  2(l+a:2)*^o 

1 
2(1 +x2) 

Substituting  this  in  (i)  gives 

J ^00 
e~^'dh  in  the  first  mem- 
0 

ber  is  equal  to  f.     Hence  ^'^  =  T,  and  from  (g) 


which  is  equation  (72). 


T  —  ^'^ 

2r 


182  APPENDIX 

D.  Evaluation  of  the  Probability  Integral.     (Supple- 
mentary to  Art.  55.)  —  The  value  of  the  integral 

y^^C^'e-^'dx  (76) 

may  be  found  when  AX  <  1  by  developing  e~^^  into  a 
series  and  integrating  the  terms  separately.  By  Mae- 
laurin's  theorem, 


/^henc 

3e 

=  1- 

^^+21- 

'3! 

4! 

"*i 

rhx 

1         6 

-^dx  - 

=  AZ 

{hxy 

1    1 

{hxy 

1 

{hxy 

0 

3 

'2! 

5 

3! 

7      ' 

(i) 


This  series  converges  rapidly  for  values  of  AZ  less  than  or 
equal  to  unity,  and  may  therefore  be  employed  in  the 
calculation  of  Y  in  this  case.  When  hX>ly  however,  it 
is  divergent.     We  may  write 

/^-^''^^  =/[-  i]-  2  --^'^-1  =/[-  il'^p- 

Integrating  successively  by  parts,  i^ 


= 

2x            2^   x"- 

=  • 

2x      4.7!^     4-^   x' 

=  - 

2  .T      4  .T^       8  x^         8 

5  Ce-^ 
J   x' 

dx^' 

L     2x     4:x^     8.1 


APPENDIX  183 

3     .   3-5 


16  a: 

3'5' 
32  a: 


7.3>5.7>9         1  ... 

"+     64  x^^  i  ^^^ 

0  */0  *^AZ 

(See  Note  (7.)  The  value  of  the  integral  in  this  last  ex- 
pression can  be  found  by  applying  the  limits  hX  and 
00  to  the  successive  terms  of  (k),  giving 


JhX  I  9  h 

+ 


hx  l2hX     4:{hXy 

3     1  15      1 


8  {hxy    16  {hxy 


•],       (m) 


which  converges  rapidly  when  hX>\.  Equation  (/) 
will  now  give  values  for  the  integral  appearing  in  Y  for 
this  case. 

Therefore,  for  }iX<\,  use  series  (j) ;  for  hX>\,  use 
series  (m)  substituted  in  (/).  (76)  will  then  yield  the 
values  of  the  probability  integral  desired.  Let  the 
student  verify  these  calculations  for,  say,  hX  =  ^  and 
hX  =  2. 

E.  Outline  of  Another  Method  for  Probable  Errors 
of  Adjusted  Values.  (Supplementary  to  Art.  62.)  — The 
method  referred  to  at  the  end  of  Art.  62  is  given  here 


184  APPENDIX 

without  proof.     (See  Merriman,  Method  of  Least  Squares, 
Art.  74.) 

Let  the  normal  equations  found  from  the  observation 
equations  be 

Aiwii  +  Bim2  4-  •  •  •  +  Rirrii  =  Ki, 
A2mi  +  B2m2  +  •  •  •  +  i^2^z  =  K2, 


AiMi  +  Bim2  +  •  •  •  +  Rirrii  =  Kj, 

in  which  the  quantities  K  take  the  place  of  Hiaws),  etc., 
in  equations  (108).  Let  the  literal  form  of  these  quanti- 
ties K  be  preserved  throughout  the  solution  of  the  normal 
equations,  which  will  then  yield 

mi  =  aiKi-{-l3,K2-h"'+\Kt, 
TTh  =  a2Ki  +  A/^2  +  •••  +  \Ki, 


^^=aiKi  +  0,K2  +  "-+\Kj 


(n) 


the  quantities  a,  /3,  •••,  X  being  numerical  coefficients  of 
the  literal  quantities  K. 

It  may  then  be  shown  that  the  weight  of  Wi  is  — ,  that  of 

Oil 

m2  is  — ,  that  of  ma  is  — ,  etc.     That  is,  the  weight  of  any 

^2  73 

most  probable  value  m^  is  the  reciprocal  of  the  coefficient 
of  the  absolute  term  Kp  of  the  normal  equation  cor- 
responding to  r)ip  as  Kp  appears  in  the  solution  (n)  for  rUp. 
The  weights  of  all  the  most  probable  values  mi,  m2,  •••, 
mi  are  thus  calculated.  The  probable  error  of  an  obser- 
vation of  unit  weight  is  given  by  (107),  or,  as  remarked 
in  Art.  62,  may  be  already  known  from  experience.     The 


APPENDIX  185 

probable  error  of  each  m  may  therefore  now  be  found  by 
dividing  this  standard  probable  error  by  the  square  root 
of  the  weight  of  m  (92),  as  above  determined. 

F.  Collection  of  Important  Definitions,  Theorems, 
Rules  and  Formulas  for  Convenient  Reference. 

DEFINITIONS 

Error.  —  The  result  of  a  measurement  minus  the  true 
value  of  the  quantity  measured.     (Art.  7.) 

Residual.  —  The  result  of  a  measurement  minus  the 
most  probable  value  of  the  quantity,  as  derived  from  a 
series  of  measurements.     (Art.  7.) 

Most  Probable  Value.  —  A  calculated  value  of  an  un- 
known quantity,  based  upon  the  results  of  measurements, 
such  that  the  residuals  arising  therefrom  will  be  most 
nearly  in  accord  with  the  normal  error  distribution. 
(Arts.  7,  29.) 

Adjustment.  —  The  process  of  obtaining  from  the  results 
of  measurements  the  most  probable  values  of  the  unknown 
quantities  sought.     (Chap.  V.) 

Observation  Equation.  —  An  equation,  in  general  only 
approximately  true,  connecting  one  or  more  unknown 
quantities,  or  functions  of  them,  with  the  result  of  a 
measurement.     (Art.  31.) 

Normal  Equation.  —  An  equation,  in  general  one  of  a 
set  of  simultaneous  equations,  whose  solution  gives  the 
most  probable  values  of  the  unknowns  involved  in  the 
observation  equations.     (Art.  33.) 


186  APPENDIX 

Equation  of  Condition.  —  An  equation  expressing  a  theo- 
retical condition  which  must  be  exactly  satisfied  by  the  cal- 
culated most  probable  values  of  the  unknowns.     (Art.  40.) 

Empirical  Formula.  —  A  formula  expressing  a  relation 
between  variables,  whose  mathematical  form  is  inferred 
from  the  results  of  experience  or  experiment,  and  which 
is  not  deduced  theoretically.     (Art.  42.) 

Weights.  —  Numbers  assigned  to  observations,  or  to 
the  adjusted  values  of  unknowns,  representing  the  relative 
degrees  of  confidence  which  the  respective  observations 
or  values  are  supposed  to  merit.     (Art.  47.) 

Weighted  Mean.  —  The  most  probable  value  of  a  single 
unknown  quantity  obtained  by  multiplying  each  obser- 
vation upon  that  quantity  by  its  weight,  adding  the  prod- 
ucts, and  dividing  by  the  sum  of  the  weights.     (Art.  48.) 

Probable  Error.  —  A  theoretical  quantity  e,  so  related 
to  the  precision  of  a  system  of  observations,  that  the 
probability  of  the  error  of  any  observation  or  adjusted 
value  being  numerically  less  than  e  is  equal  to  the  proba- 
bility of  its  being  numerically  greater.     (Art.  58.) 

RULES    AND   THEOREMS 

1.  Principle  of  Least  Squares.  —  (a)  The  most  prob- 
able value  of  a  measured  quantity  that  can  be  deduced 
from  a  series  of  direct  observations,  made  with  equal  care 
and  skill,  is  that  for  which  the  sum  of  the  squares  of  the 
residuals  is  a  minimum.     (Art.  29.) 

(b)  The  most  probable  value  of  an  unknown  quantity 
that  can  be  deduced  from  a  set  of  observations  upon  one 


APPENDIX  187 

of  its  functions  is  that  for  which  the  sum  of  the  squares  of 
the  residuals  is  a  minimum.     (Art.  31.) 

(c)  The  most  probable  values  of  unknown  quantities 
connected  by  observation  equations  are  those  for  which 
the  sum  of  the  squares  of  the  residuals  of  those  equations 
is  a  minimum.     (Art.  33.) 

{d)  The  most  probable  values  of  unknown  quantities 
connected  by  weighted  observation  equations  are  those  for 
which  the  sum  of  the  weighted  squares  of  the  residuals  is  a 
minimum.     (Art.  52.) 

2.  Rules  for  Adjusting  Observation  Equations  of  the 
First  Degree.  —  (a)  Write  the  expression  for  the  residual 
corresponding  to  each  observation  equation,  multiply  it 
by  the  coefficient  of  the  first  unknown,  in  that  expression, 
add  the  products,  and  equate  their  sum  to  zero.  The 
result  is  the  normal  equation  pertaining  to  the  said  first 
unknown.  Do  likewise  for  each  of  the  other  unknowns. 
Then  solve  the  normal  equations  thus  formed  for  the 
desired  most  probable  values  of  the  unknowns.     (Art.  34.) 

(6)  In  the  case  of  weighted  observation  equations,  after 
multiplying  the  residual  by  the  coefficient  of  the  unknown, 
multiply  again  by  the  weight  of  the  corresponding  obser- 
vation ;  then  add  and  proceed  as  above  stated.     (Art.  48.) 

3.  Weight  and  Precision  Index.  —  The  weights  of  ob- 
servations are  directly  proportional  to  the  squares  of  their 
precision  indices.     (Art.  51.) 

4.  Weight  and  Probable  Error.  —  The  weights  of  obser- 
vations are  inversely  proportional  to  the  squares  of  their 
probable  errors.     (Art.  59.) 


188  APPENDIX 

FORMULAS 

1.  The  Error  Equation.     (Art.  54.) 

y  =  ^e-''^\  (74) 

2.  Formulas  for  the  Precision  Index.     (Arts.  57,  59.) 
(a)  For  observations  of  equal  precision,  standard  for- 
mula, I 

(6)  For  weighted  observations,  standard  formula. 


^2^7^)-  (^^) 


(c)  Peters'    formula,    disregarding   signs   of   residuals, 
observations  not  weighted, 


j^^^M^-ll^  (82) 

>/7r2/3 

3.   Formulas  for  the  Probable  Errors  of  Observations  in 
Terms  of  Residuals.     (Arts.  58,  59.) 

(a)  Probable  error  of  single  observation,  no  weights, 

standard  formula,  , — 

€  =  0.6745. /-^^-  (88) 

\  M—  1 

(6)  With  weights  assigned,  probable  error  of  single  ob- 
servation of  unit  weight,  standard. 


=  0.6745./^^^^.  (94) 

\  n  —  1 


APPENDIX  189 

(c)  For  an  observation  of  unit  weight,  there  being  I  un- 
known quantities,  standard, 

e  =  0.6746./!^.  (107) 

{d)  Peters'  formulas  corresponding  to  the  above  (a), 
(6)  and  (c),  disregarding  signs  of  residuals, 

€  =  0.8453 ^ 


€  =  0.8453-5dM=.  (95) 


€  =  0.8453 


Vn(n-l) 
Mn  - 1) 


(e)  Simplified  Peters'  formulas  corresponding  to  the 
above  (a)  and  (6),  adapted  to  approximate  calculation 
when  n  is  large,  disregarding  signs  of  residuals, 

€  =  0.85^.  (90) 

n 

,  =  0.85?^^.  (96) 

n 

4.  Formulas  for  Probable  Errors  of  Functions  of  Quanti- 
ties, in  Terms  of  Probable  Errors  of  Quantities  Themselves, 
(Art.  61.) 

(a)  Function  Q  of  a  single  quantity  q, 

E  =  ^e.  (97) 

dq 


190  APPENDIX 

(6)  Function  Q  of  several  quantities,  qi,  q2,  •••,  Qi, 

--^/(S^■^(S)'•■■+■■■H^■•■■■  <-'■ 

(c)  Function  Q  =  i^i^i  +  ^292  +  •••  +  KiQi, 

E  =  'JK1W  +  K2W+  •••+K,'€;2.  (103) 

(c?)  Function  q  =  Kqi''q2^  -"  q{, 

£  =  ^(^)V+(^J,^+...+(5V.        (104) 

5.   Formulas  for  Probable  Errors  of  Adjusted   Values. 
(Art.  62.) 

(a)  For  the  arithmetical  mean,  standard, 


0.6745  J—^^.  (105) 

\n(n  — 1) 


(n-1) 
(6)  For  the  weighted  mean,  standard, 


.„„  =  0.6745  J-^(^.  (106) 

\  (n  —  l)Si/; 

(c)  Peters'   formulas    corresponding  to  the  above  (a) 

and  (6)  (Ex.  10,  Art.  64), 


0.8453 


0.8453 


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